K11a161

K11a160

K11a162

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a161 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a161/ThurstonBennequinNumber Hyperbolic Volume 11.0126 A-Polynomial See Data:K11a161/A-polynomial

[edit Notes for K11a161's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a161's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+8t^{2}-12t+13-12t^{-1}+8t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-4z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 57, -4 }
Jones polynomial	$q^{2}-2q+4-6q^{-1}+7q^{-2}-8q^{-3}+9q^{-4}-7q^{-5}+6q^{-6}-4q^{-7}+2q^{-8}-q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{8}-2a^{8}+2z^{4}a^{6}+6z^{2}a^{6}+3a^{6}-z^{6}a^{4}-3z^{4}a^{4}-z^{2}a^{4}+a^{4}-z^{6}a^{2}-4z^{4}a^{2}-5z^{2}a^{2}-3a^{2}+z^{4}+3z^{2}+2$
Kauffman polynomial (db, data sources)	$z^{3}a^{11}-za^{11}+2z^{4}a^{10}-z^{2}a^{10}+3z^{5}a^{9}-2z^{3}a^{9}+za^{9}+4z^{6}a^{8}-6z^{4}a^{8}+5z^{2}a^{8}-2a^{8}+4z^{7}a^{7}-6z^{5}a^{7}-z^{3}a^{7}+2za^{7}+4z^{8}a^{6}-9z^{6}a^{6}+z^{4}a^{6}+4z^{2}a^{6}-3a^{6}+3z^{9}a^{5}-8z^{7}a^{5}+3z^{5}a^{5}-2z^{3}a^{5}+2za^{5}+z^{10}a^{4}+2z^{8}a^{4}-21z^{6}a^{4}+29z^{4}a^{4}-13z^{2}a^{4}+a^{4}+5z^{9}a^{3}-23z^{7}a^{3}+31z^{5}a^{3}-16z^{3}a^{3}+5za^{3}+z^{10}a^{2}-z^{8}a^{2}-14z^{6}a^{2}+32z^{4}a^{2}-20z^{2}a^{2}+3a^{2}+2z^{9}a-11z^{7}a+19z^{5}a-12z^{3}a+3za+z^{8}-6z^{6}+12z^{4}-9z^{2}+2$
The A2 invariant	Data:K11a161/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a161/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a161 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["K11a161"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+8t^{2}-12t+13-12t^{-1}+8t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 57, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{8}-2a^{8}+2z^{4}a^{6}+6z^{2}a^{6}+3a^{6}-z^{6}a^{4}-3z^{4}a^{4}-z^{2}a^{4}+a^{4}-z^{6}a^{2}-4z^{4}a^{2}-5z^{2}a^{2}-3a^{2}+z^{4}+3z^{2}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{11}-za^{11}+2z^{4}a^{10}-z^{2}a^{10}+3z^{5}a^{9}-2z^{3}a^{9}+za^{9}+4z^{6}a^{8}-6z^{4}a^{8}+5z^{2}a^{8}-2a^{8}+4z^{7}a^{7}-6z^{5}a^{7}-z^{3}a^{7}+2za^{7}+4z^{8}a^{6}-9z^{6}a^{6}+z^{4}a^{6}+4z^{2}a^{6}-3a^{6}+3z^{9}a^{5}-8z^{7}a^{5}+3z^{5}a^{5}-2z^{3}a^{5}+2za^{5}+z^{10}a^{4}+2z^{8}a^{4}-21z^{6}a^{4}+29z^{4}a^{4}-13z^{2}a^{4}+a^{4}+5z^{9}a^{3}-23z^{7}a^{3}+31z^{5}a^{3}-16z^{3}a^{3}+5za^{3}+z^{10}a^{2}-z^{8}a^{2}-14z^{6}a^{2}+32z^{4}a^{2}-20z^{2}a^{2}+3a^{2}+2z^{9}a-11z^{7}a+19z^{5}a-12z^{3}a+3za+z^{8}-6z^{6}+12z^{4}-9z^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_14, K11n2,}

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["K11a161"];

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]=

{ $-2t^{3}+8t^{2}-12t+13-12t^{-1}+8t^{-2}-2t^{-3}$, $q^{2}-2q+4-6q^{-1}+7q^{-2}-8q^{-3}+9q^{-4}-7q^{-5}+6q^{-6}-4q^{-7}+2q^{-8}-q^{-9}$ }

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_14, K11n2,}

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:

(2, -7)

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 $8$ $-56$ $32$ ${\frac {748}{3}}$ ${\frac {140}{3}}$ $-448$ $-{\frac {3536}{3}}$ $-{\frac {512}{3}}$ $-216$ ${\frac {256}{3}}$ $1568$ ${\frac {5984}{3}}$ ${\frac {1120}{3}}$ ${\frac {97111}{15}}$ $-{\frac {668}{5}}$ ${\frac {128644}{45}}$ ${\frac {1049}{9}}$ ${\frac {5431}{15}}$

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 $s=$-4 is the signature of K11a161. 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 5                       1 1 3                     1   -1 1                   3 1   2 -1                 3 1     -2 -3               4 3       1 -5             5 4         -1 -7           4 3           1 -9         3 5             2 -11       3 4               -1 -13     1 3                 2 -15   1 3                   -2 -17   1                     1 -19 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a160 K11a162