K11a61

K11a60

K11a62

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a61 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X16,6,17,5 X10,8,11,7 X2,9,3,10 X18,12,19,11 X20,14,21,13 X6,16,7,15 X22,18,1,17 X14,20,15,19 X12,22,13,21
Gauss code	1, -5, 2, -1, 3, -8, 4, -2, 5, -4, 6, -11, 7, -10, 8, -3, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 8 16 10 2 18 20 6 22 14 12

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:K11a61/ThurstonBennequinNumber Hyperbolic Volume 14.3183 A-Polynomial See Data:K11a61/A-polynomial

[edit Notes for K11a61'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 K11a61's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-6t^{2}+26t-39+26t^{-1}-6t^{-2}$
Conway polynomial	$-6z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 103, 2 }
Jones polynomial	$-q^{10}+3q^{9}-6q^{8}+10q^{7}-14q^{6}+16q^{5}-16q^{4}+15q^{3}-11q^{2}+7q-3+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-3z^{4}a^{-4}-2z^{4}a^{-6}+2z^{2}a^{-2}-3z^{2}a^{-4}-z^{2}a^{-6}+3z^{2}a^{-8}+z^{2}+2a^{-2}-a^{-4}-a^{-6}+2a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+4z^{9}a^{-5}+7z^{9}a^{-7}+3z^{9}a^{-9}+7z^{8}a^{-4}+11z^{8}a^{-6}+7z^{8}a^{-8}+3z^{8}a^{-10}+8z^{7}a^{-3}+3z^{7}a^{-5}-13z^{7}a^{-7}-7z^{7}a^{-9}+z^{7}a^{-11}+6z^{6}a^{-2}-6z^{6}a^{-4}-32z^{6}a^{-6}-32z^{6}a^{-8}-12z^{6}a^{-10}+3z^{5}a^{-1}-11z^{5}a^{-3}-19z^{5}a^{-5}-3z^{5}a^{-7}-2z^{5}a^{-9}-4z^{5}a^{-11}-7z^{4}a^{-2}-5z^{4}a^{-4}+24z^{4}a^{-6}+36z^{4}a^{-8}+15z^{4}a^{-10}+z^{4}-2z^{3}a^{-1}+8z^{3}a^{-3}+13z^{3}a^{-5}+8z^{3}a^{-7}+10z^{3}a^{-9}+5z^{3}a^{-11}+6z^{2}a^{-2}+7z^{2}a^{-4}-9z^{2}a^{-6}-15z^{2}a^{-8}-6z^{2}a^{-10}-z^{2}-za^{-3}-3za^{-5}-3za^{-7}-3za^{-9}-2za^{-11}-2a^{-2}-a^{-4}+a^{-6}+2a^{-8}+a^{-10}$
The A2 invariant	Data:K11a61/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a61/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a61 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["K11a61"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-6t^{2}+26t-39+26t^{-1}-6t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 103, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{10}+3q^{9}-6q^{8}+10q^{7}-14q^{6}+16q^{5}-16q^{4}+15q^{3}-11q^{2}+7q-3+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+4z^{9}a^{-5}+7z^{9}a^{-7}+3z^{9}a^{-9}+7z^{8}a^{-4}+11z^{8}a^{-6}+7z^{8}a^{-8}+3z^{8}a^{-10}+8z^{7}a^{-3}+3z^{7}a^{-5}-13z^{7}a^{-7}-7z^{7}a^{-9}+z^{7}a^{-11}+6z^{6}a^{-2}-6z^{6}a^{-4}-32z^{6}a^{-6}-32z^{6}a^{-8}-12z^{6}a^{-10}+3z^{5}a^{-1}-11z^{5}a^{-3}-19z^{5}a^{-5}-3z^{5}a^{-7}-2z^{5}a^{-9}-4z^{5}a^{-11}-7z^{4}a^{-2}-5z^{4}a^{-4}+24z^{4}a^{-6}+36z^{4}a^{-8}+15z^{4}a^{-10}+z^{4}-2z^{3}a^{-1}+8z^{3}a^{-3}+13z^{3}a^{-5}+8z^{3}a^{-7}+10z^{3}a^{-9}+5z^{3}a^{-11}+6z^{2}a^{-2}+7z^{2}a^{-4}-9z^{2}a^{-6}-15z^{2}a^{-8}-6z^{2}a^{-10}-z^{2}-za^{-3}-3za^{-5}-3za^{-7}-3za^{-9}-2za^{-11}-2a^{-2}-a^{-4}+a^{-6}+2a^{-8}+a^{-10}$

"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["K11a61"];

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]=

{ $-6t^{2}+26t-39+26t^{-1}-6t^{-2}$, $-q^{10}+3q^{9}-6q^{8}+10q^{7}-14q^{6}+16q^{5}-16q^{4}+15q^{3}-11q^{2}+7q-3+q^{-1}$ }

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:

(2, 5)

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 $8$ $40$ $32$ ${\frac {700}{3}}$ ${\frac {188}{3}}$ $320$ ${\frac {3568}{3}}$ ${\frac {640}{3}}$ $296$ ${\frac {256}{3}}$ $800$ ${\frac {5600}{3}}$ ${\frac {1504}{3}}$ ${\frac {87871}{15}}$ $-{\frac {3788}{5}}$ ${\frac {165124}{45}}$ ${\frac {1217}{9}}$ ${\frac {8431}{15}}$

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 K11a61. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a60 K11a62