K11a104

K11a103

K11a105

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a104 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a104/ThurstonBennequinNumber Hyperbolic Volume 15.7183 A-Polynomial See Data:K11a104/A-polynomial

[edit Notes for K11a104's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $1$ Rasmussen s-Invariant 0

[edit Notes for K11a104's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+12t^{2}-29t+39-29t^{-1}+12t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 125, 0 }
Jones polynomial	$q^{6}-4q^{5}+8q^{4}-13q^{3}+18q^{2}-20q+20-17q^{-1}+13q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-z^{4}-a^{4}z^{2}+3a^{2}z^{2}-2z^{2}a^{-2}+z^{2}a^{-4}-a^{4}+2a^{2}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}+4az^{9}+8z^{9}a^{-1}+4z^{9}a^{-3}+6a^{2}z^{8}+14z^{8}a^{-2}+6z^{8}a^{-4}+14z^{8}+5a^{3}z^{7}+4az^{7}-3z^{7}a^{-1}+2z^{7}a^{-3}+4z^{7}a^{-5}+3a^{4}z^{6}-6a^{2}z^{6}-36z^{6}a^{-2}-13z^{6}a^{-4}+z^{6}a^{-6}-31z^{6}+a^{5}z^{5}-6a^{3}z^{5}-16az^{5}-21z^{5}a^{-1}-22z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+a^{2}z^{4}+26z^{4}a^{-2}+7z^{4}a^{-4}-2z^{4}a^{-6}+23z^{4}-2a^{5}z^{3}+2a^{3}z^{3}+14az^{3}+21z^{3}a^{-1}+18z^{3}a^{-3}+7z^{3}a^{-5}+3a^{4}z^{2}+3a^{2}z^{2}-7z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-5z^{2}+a^{5}z-4az-6za^{-1}-4za^{-3}-za^{-5}-a^{4}-2a^{2}$
The A2 invariant	Data:K11a104/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a104/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a104 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["K11a104"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+12t^{2}-29t+39-29t^{-1}+12t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 125, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-4q^{5}+8q^{4}-13q^{3}+18q^{2}-20q+20-17q^{-1}+13q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}+4az^{9}+8z^{9}a^{-1}+4z^{9}a^{-3}+6a^{2}z^{8}+14z^{8}a^{-2}+6z^{8}a^{-4}+14z^{8}+5a^{3}z^{7}+4az^{7}-3z^{7}a^{-1}+2z^{7}a^{-3}+4z^{7}a^{-5}+3a^{4}z^{6}-6a^{2}z^{6}-36z^{6}a^{-2}-13z^{6}a^{-4}+z^{6}a^{-6}-31z^{6}+a^{5}z^{5}-6a^{3}z^{5}-16az^{5}-21z^{5}a^{-1}-22z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+a^{2}z^{4}+26z^{4}a^{-2}+7z^{4}a^{-4}-2z^{4}a^{-6}+23z^{4}-2a^{5}z^{3}+2a^{3}z^{3}+14az^{3}+21z^{3}a^{-1}+18z^{3}a^{-3}+7z^{3}a^{-5}+3a^{4}z^{2}+3a^{2}z^{2}-7z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-5z^{2}+a^{5}z-4az-6za^{-1}-4za^{-3}-za^{-5}-a^{4}-2a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a67, K11a168,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a168,}

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["K11a104"];

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]=

{ $-2t^{3}+12t^{2}-29t+39-29t^{-1}+12t^{-2}-2t^{-3}$, $q^{6}-4q^{5}+8q^{4}-13q^{3}+18q^{2}-20q+20-17q^{-1}+13q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$ }

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]=

{K11a67, K11a168,}

In[6]:=

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

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

Out[6]=

{K11a168,}

Vassiliev invariants

V2 and V3:

(1, -1)

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 $4$ $-8$ $8$ ${\frac {62}{3}}$ ${\frac {10}{3}}$ $-32$ $-{\frac {80}{3}}$ ${\frac {64}{3}}$ $-8$ ${\frac {32}{3}}$ $32$ ${\frac {248}{3}}$ ${\frac {40}{3}}$ ${\frac {6031}{30}}$ ${\frac {1738}{15}}$ $-{\frac {1258}{45}}$ ${\frac {305}{18}}$ $-{\frac {1169}{30}}$

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 K11a104. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   5 1   4 7                 8 3     -5 5               10 5       5 3             10 8         -2 1           10 10           0 -1         8 11             3 -3       5 9               -4 -5     2 8                 6 -7   1 5                   -4 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a103 K11a105