K11n91

K11n90

K11n92

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n91 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X7,16,8,17 X9,18,10,19 X2,11,3,12 X20,13,21,14 X15,8,16,9 X17,6,18,7 X22,19,1,20 X14,21,15,22
Gauss code	1, -6, 2, -1, 3, 9, -4, 8, -5, -2, 6, -3, 7, -11, -8, 4, -9, 5, 10, -7, 11, -10
Dowker-Thistlethwaite code	4 10 12 -16 -18 2 20 -8 -6 22 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n91/ThurstonBennequinNumber Hyperbolic Volume 9.90352 A-Polynomial See Data:K11n91/A-polynomial

[edit Notes for K11n91's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 2

[edit Notes for K11n91's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+8t-13+8t^{-1}-t^{-2}$
Conway polynomial	$-z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 31, -2 }
Jones polynomial	$q^{-1}-2q^{-2}+4q^{-3}-4q^{-4}+5q^{-5}-5q^{-6}+4q^{-7}-3q^{-8}+2q^{-9}-q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$-a^{10}+2z^{2}a^{8}+2a^{8}-z^{4}a^{6}-2z^{2}a^{6}-3a^{6}+3z^{2}a^{4}+3a^{4}+z^{2}a^{2}$
Kauffman polynomial (db, data sources)	$z^{7}a^{11}-5z^{5}a^{11}+7z^{3}a^{11}-3za^{11}+2z^{8}a^{10}-10z^{6}a^{10}+14z^{4}a^{10}-6z^{2}a^{10}+a^{10}+z^{9}a^{9}-2z^{7}a^{9}-6z^{5}a^{9}+11z^{3}a^{9}-3za^{9}+4z^{8}a^{8}-18z^{6}a^{8}+23z^{4}a^{8}-11z^{2}a^{8}+2a^{8}+z^{9}a^{7}-2z^{7}a^{7}-2z^{5}a^{7}+2z^{3}a^{7}-za^{7}+2z^{8}a^{6}-8z^{6}a^{6}+13z^{4}a^{6}-12z^{2}a^{6}+3a^{6}+z^{7}a^{5}-z^{5}a^{5}-za^{5}+4z^{4}a^{4}-6z^{2}a^{4}+3a^{4}+2z^{3}a^{3}+z^{2}a^{2}$
The A2 invariant	Data:K11n91/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n91/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n91 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["K11n91"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{2}+8t-13+8t^{-1}-t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 31, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-1}-2q^{-2}+4q^{-3}-4q^{-4}+5q^{-5}-5q^{-6}+4q^{-7}-3q^{-8}+2q^{-9}-q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{10}+2z^{2}a^{8}+2a^{8}-z^{4}a^{6}-2z^{2}a^{6}-3a^{6}+3z^{2}a^{4}+3a^{4}+z^{2}a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{11}-5z^{5}a^{11}+7z^{3}a^{11}-3za^{11}+2z^{8}a^{10}-10z^{6}a^{10}+14z^{4}a^{10}-6z^{2}a^{10}+a^{10}+z^{9}a^{9}-2z^{7}a^{9}-6z^{5}a^{9}+11z^{3}a^{9}-3za^{9}+4z^{8}a^{8}-18z^{6}a^{8}+23z^{4}a^{8}-11z^{2}a^{8}+2a^{8}+z^{9}a^{7}-2z^{7}a^{7}-2z^{5}a^{7}+2z^{3}a^{7}-za^{7}+2z^{8}a^{6}-8z^{6}a^{6}+13z^{4}a^{6}-12z^{2}a^{6}+3a^{6}+z^{7}a^{5}-z^{5}a^{5}-za^{5}+4z^{4}a^{4}-6z^{2}a^{4}+3a^{4}+2z^{3}a^{3}+z^{2}a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11n91"];

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]=

{ $-t^{2}+8t-13+8t^{-1}-t^{-2}$, $q^{-1}-2q^{-2}+4q^{-3}-4q^{-4}+5q^{-5}-5q^{-6}+4q^{-7}-3q^{-8}+2q^{-9}-q^{-10}$ }

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]=

{}

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:

(4, -9)

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 $16$ $-72$ $128$ ${\frac {1208}{3}}$ ${\frac {208}{3}}$ $-1152$ $-2416$ $-416$ $-392$ ${\frac {2048}{3}}$ $2592$ ${\frac {19328}{3}}$ ${\frac {3328}{3}}$ ${\frac {219182}{15}}$ $-{\frac {1376}{5}}$ ${\frac {299408}{45}}$ ${\frac {1378}{9}}$ ${\frac {13742}{15}}$

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 $s=$-2 is the signature of K11n91. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                   1 1 -3                 2 1 -1 -5               2     2 -7             2 2     0 -9           3 2       1 -11         2 2         0 -13       2 3           -1 -15     1 2             1 -17   1 2               -1 -19   1                 1 -21 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

K11n90 K11n92