# Skein Invariants of Links and Their State Sum Models

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## Abstract

**:**

## 1. Introduction

## 2. The Skein-Theoretic Setting for the Generalized Invariants

#### 2.1. Defining $H\left[H\right]$

**Theorem**

**1**

**.**Let $H(z,a)$ denote the regular isotopy version of the Homflypt polynomial. Then there exists a unique regular isotopy invariant of classical oriented links $H\left[H\right]:\mathcal{L}\to \mathbb{Z}[z,{a}^{\pm 1},{E}^{\pm 1}]$, where $z,\phantom{\rule{0.166667em}{0ex}}a$ and E are indeterminates, defined by the following rules:

- On crossings involving different components the following mixed skein relation holds:$$H\left[H\right]\left({L}_{+}\right)-H\left[H\right]\left({L}_{-}\right)=z\phantom{\rule{0.166667em}{0ex}}H\left[H\right]\left({L}_{0}\right),$$
- For a union of r unlinked knots, ${\mathcal{K}}^{r}:={\bigsqcup}_{i=1}^{r}{K}_{i}$, with $r\ge 1$, it holds that:$$H\left[H\right]\left({\mathcal{K}}^{r}\right)={E}^{1-r}\phantom{\rule{0.166667em}{0ex}}H\left({\mathcal{K}}^{r}\right).$$

- (H1)
- For ${L}_{+}$, ${L}_{-}$, ${L}_{0}$ an oriented Conway triple, the following skein relation holds:$$H\left({L}_{+}\right)-H\left({L}_{-}\right)=z\phantom{\rule{0.166667em}{0ex}}H\left({L}_{0}\right),$$
- (H2)
- The indeterminate a is the positive curl value for H:
- (H3)
- On the standard unknot:$$R(\u25ef)=1.$$

- (H4)
- For a diagram of the unknot, U, H is evaluated by taking:$$H\left(U\right)={a}^{wr\left(U\right)},$$
- (H5)
- H being the Homflypt polynomial, it is multiplicative on a union of unlinked knots, ${\mathcal{K}}^{r}:={\bigsqcup}_{i=1}^{r}{K}_{i}$. Namely, for $\eta :=\frac{a-{a}^{-1}}{z}$ we have:$$H\left({\mathcal{K}}^{r}\right)={\eta}^{r-1}{\Pi}_{i=1}^{r}H\left({K}_{i}\right).$$

#### 2.2. Defining $D\left[D\right]$ and $K\left[K\right]$

**Theorem**

**2**

**.**Let $D(z,a)$ denote the regular isotopy version of the Dubrovnik polynomial. Then there exists a unique regular isotopy invariant of classical unoriented links $D\left[D\right]:{\mathcal{L}}^{u}\to \mathbb{Z}[z,{a}^{\pm 1},{E}^{\pm 1}]$, where $z,\phantom{\rule{0.166667em}{0ex}}a$ and E are indeterminates, defined by the following rules:

- On crossings involving different components the following skein relation holds:$$D\left[D\right]\left({L}_{+}\right)-D\left[D\right]\left({L}_{-}\right)=z\phantom{\rule{0.166667em}{0ex}}\left(D\left[D\right]\left({L}_{0}\right)-D\left[D\right]\left({L}_{\infty}\right)\right),$$
- For a union of r unlinked knots in ${\mathcal{L}}^{u}$, ${\mathcal{K}}^{r}:={\bigsqcup}_{i=1}^{r}{K}_{i}$, with $r\ge 1$, it holds that:$$D\left[D\right]\left({\mathcal{K}}^{r}\right)={E}^{1-r}\phantom{\rule{0.166667em}{0ex}}D\left({\mathcal{K}}^{r}\right).$$

- (D1)
- For ${L}_{+}$, ${L}_{-}$, ${L}_{0}$, ${L}_{\infty}$ an unoriented Conway quadruple, the following skein relation holds:$$D\left({L}_{+}\right)-D\left({L}_{-}\right)=z\phantom{\rule{0.166667em}{0ex}}\left(D\left({L}_{0}\right)-D\left({L}_{\infty}\right)\right),$$
- (D2)
- The indeterminate a is the positive type curl value for D:
- (D3)
- On the standard unknot:$$D(\u25ef)=1.$$

- (D4)
- For a diagram of the unknot, U, D is evaluated by taking$$D\left(U\right)={a}^{wr\left(U\right)},$$
- (D5)
- D, being the Dubrovnik polynomial, it is multiplicative on a union of unlinked knots, ${\mathcal{K}}^{r}:={\bigsqcup}_{i=1}^{r}{K}_{i}$. Namely, for $\delta :=\frac{a-{a}^{-1}}{z}+1$ we have:$$D\left({\mathcal{K}}^{r}\right)={\delta}^{r-1}{\Pi}_{i=1}^{r}D\left({K}_{i}\right).$$

**Theorem**

**3**

**.**Let $K(z,a)$ denote the regular isotopy version of the Kauffman polynomial. Then there exists a unique regular isotopy invariant of classical unoriented links $K\left[K\right]:{\mathcal{L}}^{u}\to \mathbb{Z}[z,{a}^{\pm 1},{E}^{\pm 1}]$, where $z,\phantom{\rule{0.166667em}{0ex}}a$ and E are indeterminates, defined by the following rules:

- On crossings involving different components the following skein relation holds:$$K\left[K\right]\left({L}_{+}\right)+K\left[K\right]\left({L}_{-}\right)=z\phantom{\rule{0.166667em}{0ex}}\left(K\left[K\right]\left({L}_{0}\right)+K\left[K\right]\left({L}_{\infty}\right)\right),$$
- For a union of r unlinked knots in ${\mathcal{L}}^{u}$, ${\mathcal{K}}^{r}:={\bigsqcup}_{i=1}^{r}{K}_{i}$, with $r\ge 1$, it holds that:$$K\left[K\right]\left({\mathcal{K}}^{r}\right)={E}^{1-r}\phantom{\rule{0.166667em}{0ex}}K\left({\mathcal{K}}^{r}\right).$$

- (K1)
- For ${L}_{+}$, ${L}_{-}$, ${L}_{0}$, ${L}_{\infty}$ an unoriented Conway quadruple, the following skein relation holds:$$K\left({L}_{+}\right)+K\left({L}_{-}\right)=z\phantom{\rule{0.166667em}{0ex}}\left(K\left({L}_{0}\right)+K\left({L}_{\infty}\right)\right),$$
- (K2)
- The indeterminate a is the positive type curl value for K:
- (K3)
- On the standard unknot:$$K(\u25ef)=1.$$

- (K4)
- For a diagram of the unknot, U, K is evaluated by taking$$K\left(U\right)={a}^{wr\left(U\right)},$$
- (K5)
- K, being the Kauffman polynomial, it is multiplicative on a union of unlinked knots, ${\mathcal{K}}^{r}:={\bigsqcup}_{i=1}^{r}{K}_{i}$. Namely, for $\gamma :=\frac{a+{a}^{-1}}{z}-1$ we have:$$K\left({\mathcal{K}}^{r}\right)={\gamma}^{r-1}{\Pi}_{i=1}^{r}K\left({K}_{i}\right).$$

#### 2.3. Terminology and Notations

#### 2.4. Computing Algorithm for the Generalized Invariants

- (Diagrammatic level) Make L generic by choosing an order for its components and a basepoint and a direction on each component. Start from the basepoint of the first component and go along it in the chosen direction. When arriving at a mixed crossing for the first time along an under-arc we switch it by the mixed skein relation, so that we pass by the mixed crossing along the over-arc. At the same time we smooth the mixed crossing, obtaining a new diagram in which the two components of the crossing merge into one. We repeat for all mixed crossings of the first component. Among all resulting diagrams there is only one with the same number of crossings and the same number of components as the initial diagram and in this one the first component gets unlinked from the rest and lies above all of them. The other resulting diagrams have one less crossing and have the first component fused together with some other component. We proceed similarly with the second component switching all its mixed crossings except for crossings involving the first component. In the end the second component gets unliked from all the rest and lies below the first one and above all others in the maximal crossing diagram, while we also obtain diagrams containing mergings of the second component with others (except component one). We continue in the same manner with all components in order and we also apply this procedure to all product diagrams coming from smoothings of mixed crossings. In the end we obtain the unlinked version of L plus a number of links ℓ with unlinked components resulting from the mergings of different components.
- (Computational level) On the level of the generalized invariant, Rule (1) of Theorems 1–3 tells us how the switching of mixed crossings is controlled. After all applications of the mixed skein relation we obtain a linear sum of the values of the generalized invariant on all the resulting links ℓ with unlinked components. The evaluation of the generalized invariant on each ℓ reduces to the evaluation of the corresponding basic invariant by Rule (2) of Theorems 1–3.

#### 2.5. Sketching the Proof of Theorems 1–3

## 3. Translations to Ambient Isotopy

#### 3.1. Normalization of $H\left[H\right]$

**Theorem**

**4**

**.**Let $P(z,a)$ denote the Homflypt polynomial. Then there exists a unique ambient isotopy invariant of classical oriented links $P\left[P\right]:\mathcal{L}\to \mathbb{Z}[z,{a}^{\pm 1},{E}^{\pm 1}]$ defined by the following rules:

- On crossings involving different components the following skein relation holds:$$a\phantom{\rule{0.166667em}{0ex}}P\left[P\right]\left({L}_{+}\right)-{a}^{-1}\phantom{\rule{0.166667em}{0ex}}P\left[P\right]\left({L}_{-}\right)=z\phantom{\rule{0.166667em}{0ex}}P\left[P\right]\left({L}_{0}\right),$$
- For ${\mathcal{K}}^{r}:={\bigsqcup}_{i=1}^{r}{K}_{i}$, a union of r unlinked knots, with $r\ge 1$, it holds that:$$P\left[P\right]\left({\mathcal{K}}^{r}\right)={E}^{1-r}\phantom{\rule{0.166667em}{0ex}}P\left({\mathcal{K}}^{r}\right).$$

**Remark**

**1.**

#### 3.2. Normalization of $D\left[D\right]$ and $K\left[K\right]$

## 4. Closed Combinatorial Formuli for the Generalized Invariants

#### 4.1. A Closed Formula for $H\left[H\right]$

**Theorem**

**5**

**.**Let L be an oriented link with n components. Then

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

#### 4.2. An Example

**Example**

**1.**

#### 4.3. Closed Formuli for $D\left[D\right]$ and $K\left[K\right]$

**Theorem**

**6**

**.**Let L be an unoriented link with n components. Then

**Theorem**

**7**

**.**Let L be an unoriented link with n components. Then

**Remark**

**5.**

## 5. State Sum Models

**Definition**

**1.**

#### 5.1. The Skein Template Algorithm

- When traveling through a smoothing, label it by a dot and a connector indicating the place of first passage as shown in Figure 3 and exemplified in Figure 6 and Figure 7. At a smoothing, assign to the smoothing a vertex weight of $+z$ or $-z$ (the weights are indicated in Figure 8).We clarify these steps with two examples, the Hopf link and the Whitehead link. See Figure 6 and Figure 7. In these figures, for Step 1 we start at the edge with index 1 and meet a mixed crossing at its under-arc, switching it for one diagram and smoothing it for another. We walk past the smoothing, placing a dot and a connector.
- When meeting a mixed over-crossing, circle the crossing (Figure 5 middle) to indicate that it has been processed and continue the walk.
- When meeting a self-crossing, leave it unmarked (Figure 5 bottom) and continue the walk.
- When a closed path has been traversed in the template, choose the next lowest unused template index and start a new walk. Follow the previous instructions for this walk, only labeling smoothings or circling crossings that have not already been so marked.
- When all paths have been traversed, and the pre-state has no remaining un-processed mixed crossing, the pre-state $\widehat{S}$ is now a state S for L. When we have a state S, it is not hard to see that it consists in an unlinked collection of components in the form of stacks of knots as we have previously described in this paper.
- When a pre-state is finished, there will be no undecorated mixed crossings in the state. All uncircled crossings will be self-crossings and there will also be some marked smoothings. All the smoothings will have non-zero vertex weights ( z, $-z$ or 1) and the pre-state becomes a contributing state for the invariant.
- This state is evaluated by taking the product of the vertex weights and the evaluation of the invariant R on the the link underlying the state after all the decorations have been removed. The skein template process produces a link from the state that is a stack of knots. We give the details in the next section.
- The (unnormalized) invariant $H\left[R\right]$ is the sum over all the evaluations of these states obtained by applying the skein-template algorithm. We will denote this sum by $Z\left[R\right]\left(L\right)$ for a given link L and justify in the discussion below that it is indeed equal to the previously defined $H\left[R\right]\left(L\right)$.

#### 5.2. The State Summation

**Definition**

**2.**

**Remark**

**6.**

#### 5.3. Connection of the State Sum with Skein Calculation

- its two arcs lie on separate components of the given diagram;
- the walker for the skein process always switches a mixed crossing that the walker approaches as an under-crossing, and never switches a crossing that it approaches as an over-crossing;
- in expanding the crossing, the walker is shifted along according to the illustrations in Figure 4.

**Theorem**

**8.**

**Proof.**

**Remark**

**7.**

**Example**

**2.**

**Remark**

**8.**

## 6. Double State Summations

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Remark**

**9.**

**Remark**

**10.**

## 7. Statistical Mechanics and Double State Summations

## 8. Discussing Applications

- Reconnection (in vortices). In a knotted vortex in a fluid or plasma (for example in solar flares) [49] one has a cascade of changes in the vortex topology as strands of the vortex undergo reconnection. The process goes on until the vortex has degenerated into a disjoint union of unknotted simpler vortices. This cascade or hierarchy of interactions is reminiscent of the way the skein template algorithm proceeds to produce unlinks. Studying reconnection in vortices may be facilitated by making a statistical mechanics summation related to the cascade. Such a summation will be analogous the state summations we have described here.
- In DNA, strand switching using topoisomerase of types I and II is vital for the structure of DNA replication [50]. The mixed interaction of topological change and physical evolution of the molecules in vitro may benefit from a mixed state summation that averages quantities respecting the hierarchy of interactions.
- Remarkably, the process of separation and evaluation that we have described here is analogous to proposed processing of Kinetoplast DNA [51] where there are huge links of DNA circles and these must undergo processes that both unlink them from one another and produce new copies for each circle of DNA. The double-tiered structure of DNA replication for the Kinetoplast chainmail DNA appears to be related to the mathematical patterns of our double state summations. If the reader examines the Wiki on Kinetoplast DNA, he/she will note that Topoisomerase II figures crucially in the self-replication [52].
- We wondered whether we could have physical situations that would have the kind of a mixture that is implicit in this state summation, where the initial skein template state sum yields a sum over R-evaluations, and R may itself have a state summation structure. One possible example in the physical world is a normal statistical mechanical situation, where one can have multiple types of materials, all present together, each having different energetic properties. This can lead to a mixed partition function, possibly not quite ordered in the fashion of our algorithm. This would involve a physical hierarchy of interactions so that there would be a double (or multiple) tier resulting from that hierarchy.
- Mixed state models can occur in physical situations when we work with systems of systems. There are many examples of this multiple-tier situation in systems physical and biological. We look for situations where a double state sum would yield new information. For example, in a quantum Hall system [53], the state of the system is in its quasi-particles, but each quasi-particle is itself a vortex of electrons related to a magnetic field line. So the quasi-particles are themselves localized physical systems. Some of this is summarized in the Laughlin wave function for quantum Hall [53]. This is not a simple situation, but a very significant one. There should be other important examples.

## 9. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Kauffman, L.H.; Lambropoulou, S.
Skein Invariants of Links and Their State Sum Models. *Symmetry* **2017**, *9*, 226.
https://doi.org/10.3390/sym9100226

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Kauffman LH, Lambropoulou S.
Skein Invariants of Links and Their State Sum Models. *Symmetry*. 2017; 9(10):226.
https://doi.org/10.3390/sym9100226

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Kauffman, Louis H., and Sofia Lambropoulou.
2017. "Skein Invariants of Links and Their State Sum Models" *Symmetry* 9, no. 10: 226.
https://doi.org/10.3390/sym9100226