K11a101

K11a100

K11a102

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a101 at Knotilus!

Knot presentations

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

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:K11a101/ThurstonBennequinNumber Hyperbolic Volume 18.521 A-Polynomial See Data:K11a101/A-polynomial

[edit Notes for K11a101's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for K11a101's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-16t^{2}+39t-51+39t^{-1}-16t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 167, 2 }
Jones polynomial	$q^{9}-5q^{8}+11q^{7}-18q^{6}+24q^{5}-27q^{4}+27q^{3}-23q^{2}+17q-9+4q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+2z^{6}a^{-4}+z^{4}a^{-2}+5z^{4}a^{-4}-3z^{4}a^{-6}-z^{4}+z^{2}a^{-2}+5z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}-z^{2}+a^{-2}+a^{-4}-a^{-6}$
Kauffman polynomial (db, data sources)	$3z^{10}a^{-4}+3z^{10}a^{-6}+9z^{9}a^{-3}+18z^{9}a^{-5}+9z^{9}a^{-7}+11z^{8}a^{-2}+20z^{8}a^{-4}+19z^{8}a^{-6}+10z^{8}a^{-8}+8z^{7}a^{-1}-4z^{7}a^{-3}-26z^{7}a^{-5}-9z^{7}a^{-7}+5z^{7}a^{-9}-15z^{6}a^{-2}-54z^{6}a^{-4}-57z^{6}a^{-6}-21z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-10z^{5}a^{-1}-12z^{5}a^{-3}-5z^{5}a^{-5}-13z^{5}a^{-7}-9z^{5}a^{-9}+9z^{4}a^{-2}+44z^{4}a^{-4}+43z^{4}a^{-6}+12z^{4}a^{-8}-z^{4}a^{-10}-5z^{4}-az^{3}+5z^{3}a^{-1}+13z^{3}a^{-3}+16z^{3}a^{-5}+13z^{3}a^{-7}+4z^{3}a^{-9}-z^{2}a^{-2}-13z^{2}a^{-4}-12z^{2}a^{-6}-2z^{2}a^{-8}+2z^{2}-za^{-1}-3za^{-3}-5za^{-5}-3za^{-7}-a^{-2}+a^{-4}+a^{-6}$
The A2 invariant	Data:K11a101/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a101/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a101 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["K11a101"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{3}-16t^{2}+39t-51+39t^{-1}-16t^{-2}+3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{6}+2z^{4}+2z^{2}+1$

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]=

{ 167, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{9}-5q^{8}+11q^{7}-18q^{6}+24q^{5}-27q^{4}+27q^{3}-23q^{2}+17q-9+4q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{10}a^{-4}+3z^{10}a^{-6}+9z^{9}a^{-3}+18z^{9}a^{-5}+9z^{9}a^{-7}+11z^{8}a^{-2}+20z^{8}a^{-4}+19z^{8}a^{-6}+10z^{8}a^{-8}+8z^{7}a^{-1}-4z^{7}a^{-3}-26z^{7}a^{-5}-9z^{7}a^{-7}+5z^{7}a^{-9}-15z^{6}a^{-2}-54z^{6}a^{-4}-57z^{6}a^{-6}-21z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-10z^{5}a^{-1}-12z^{5}a^{-3}-5z^{5}a^{-5}-13z^{5}a^{-7}-9z^{5}a^{-9}+9z^{4}a^{-2}+44z^{4}a^{-4}+43z^{4}a^{-6}+12z^{4}a^{-8}-z^{4}a^{-10}-5z^{4}-az^{3}+5z^{3}a^{-1}+13z^{3}a^{-3}+16z^{3}a^{-5}+13z^{3}a^{-7}+4z^{3}a^{-9}-z^{2}a^{-2}-13z^{2}a^{-4}-12z^{2}a^{-6}-2z^{2}a^{-8}+2z^{2}-za^{-1}-3za^{-3}-5za^{-5}-3za^{-7}-a^{-2}+a^{-4}+a^{-6}$

"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["K11a101"];

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]=

{ $3t^{3}-16t^{2}+39t-51+39t^{-1}-16t^{-2}+3t^{-3}$, $q^{9}-5q^{8}+11q^{7}-18q^{6}+24q^{5}-27q^{4}+27q^{3}-23q^{2}+17q-9+4q^{-1}-q^{-2}$ }

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, 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 $8$ $24$ $32$ ${\frac {220}{3}}$ $-{\frac {4}{3}}$ $192$ $304$ $64$ $-8$ ${\frac {256}{3}}$ $288$ ${\frac {1760}{3}}$ $-{\frac {32}{3}}$ ${\frac {18991}{15}}$ ${\frac {3116}{15}}$ ${\frac {10564}{45}}$ $-{\frac {319}{9}}$ ${\frac {271}{15}}$

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 K11a101. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     4   -4 15                   7 1   6 13                 11 4     -7 11               13 7       6 9             14 11         -3 7           13 13           0 5         10 14             4 3       7 13               -6 1     3 11                 8 -1   1 6                   -5 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a100 K11a102