K11a93

K11a92

K11a94

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a93 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X16,7,17,8 X22,10,1,9 X2,11,3,12 X20,13,21,14 X18,15,19,16 X8,17,9,18 X14,19,15,20 X6,22,7,21
Gauss code	1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -8, 10, -7, 11, -5
Dowker-Thistlethwaite code	4 10 12 16 22 2 20 18 8 14 6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a93/ThurstonBennequinNumber Hyperbolic Volume 13.1112 A-Polynomial See Data:K11a93/A-polynomial

[edit Notes for K11a93's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11a93's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+10t^{2}-21t+27-21t^{-1}+10t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 93, 0 }
Jones polynomial	$q^{4}-3q^{3}+6q^{2}-10q+13-14q^{-1}+15q^{-2}-12q^{-3}+9q^{-4}-6q^{-5}+3q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}-a^{6}+2z^{4}a^{4}+4z^{2}a^{4}+a^{4}-z^{6}a^{2}-2z^{4}a^{2}+2a^{2}-z^{6}-3z^{4}-4z^{2}-2+z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+6a^{3}z^{9}+3az^{9}+3a^{6}z^{8}+5a^{4}z^{8}+7a^{2}z^{8}+5z^{8}+a^{7}z^{7}-8a^{5}z^{7}-12a^{3}z^{7}+3az^{7}+6z^{7}a^{-1}-12a^{6}z^{6}-25a^{4}z^{6}-20a^{2}z^{6}+5z^{6}a^{-2}-2z^{6}-4a^{7}z^{5}-2a^{3}z^{5}-15az^{5}-6z^{5}a^{-1}+3z^{5}a^{-3}+14a^{6}z^{4}+26a^{4}z^{4}+11a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-7z^{4}+5a^{7}z^{3}+9a^{5}z^{3}+8a^{3}z^{3}+9az^{3}+2z^{3}a^{-1}-3z^{3}a^{-3}-5a^{6}z^{2}-9a^{4}z^{2}-a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+7z^{2}-2a^{7}z-3a^{5}z-2a^{3}z-az+za^{-1}+za^{-3}+a^{6}+a^{4}-2a^{2}-a^{-2}-2$
The A2 invariant	Data:K11a93/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a93/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a93 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a93"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-2t^{3}+10t^{2}-21t+27-21t^{-1}+10t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-2z^{4}+z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 93, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{4}-3q^{3}+6q^{2}-10q+13-14q^{-1}+15q^{-2}-12q^{-3}+9q^{-4}-6q^{-5}+3q^{-6}-q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{2}a^{6}-a^{6}+2z^{4}a^{4}+4z^{2}a^{4}+a^{4}-z^{6}a^{2}-2z^{4}a^{2}+2a^{2}-z^{6}-3z^{4}-4z^{2}-2+z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+6a^{3}z^{9}+3az^{9}+3a^{6}z^{8}+5a^{4}z^{8}+7a^{2}z^{8}+5z^{8}+a^{7}z^{7}-8a^{5}z^{7}-12a^{3}z^{7}+3az^{7}+6z^{7}a^{-1}-12a^{6}z^{6}-25a^{4}z^{6}-20a^{2}z^{6}+5z^{6}a^{-2}-2z^{6}-4a^{7}z^{5}-2a^{3}z^{5}-15az^{5}-6z^{5}a^{-1}+3z^{5}a^{-3}+14a^{6}z^{4}+26a^{4}z^{4}+11a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}-7z^{4}+5a^{7}z^{3}+9a^{5}z^{3}+8a^{3}z^{3}+9az^{3}+2z^{3}a^{-1}-3z^{3}a^{-3}-5a^{6}z^{2}-9a^{4}z^{2}-a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+7z^{2}-2a^{7}z-3a^{5}z-2a^{3}z-az+za^{-1}+za^{-3}+a^{6}+a^{4}-2a^{2}-a^{-2}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_114,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a93"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $-2t^{3}+10t^{2}-21t+27-21t^{-1}+10t^{-2}-2t^{-3}$, $q^{4}-3q^{3}+6q^{2}-10q+13-14q^{-1}+15q^{-2}-12q^{-3}+9q^{-4}-6q^{-5}+3q^{-6}-q^{-7}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{10_114,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(1, -3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $4$ $-24$ $8$ ${\frac {206}{3}}$ ${\frac {58}{3}}$ $-96$ $-208$ $0$ $-56$ ${\frac {32}{3}}$ $288$ ${\frac {824}{3}}$ ${\frac {232}{3}}$ ${\frac {28831}{30}}$ $-{\frac {34}{5}}$ ${\frac {17942}{45}}$ ${\frac {641}{18}}$ ${\frac {991}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of K11a93. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                       1 1 7                     2   -2 5                   4 1   3 3                 6 2     -4 1               7 4       3 -1             8 7         -1 -3           7 6           1 -5         5 8             3 -7       4 7               -3 -9     2 5                 3 -11   1 4                   -3 -13   2                     2 -15 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a92 K11a94