K11a100

K11a99

K11a101

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a100 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a100/ThurstonBennequinNumber Hyperbolic Volume 16.1755 A-Polynomial See Data:K11a100/A-polynomial

[edit Notes for K11a100'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 K11a100's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{3}+15t^{2}-32t+41-32t^{-1}+15t^{-2}-3t^{-3}$
Conway polynomial	$-3z^{6}-3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 141, 4 }
Jones polynomial	$-q^{11}+5q^{10}-10q^{9}+15q^{8}-21q^{7}+23q^{6}-22q^{5}+19q^{4}-13q^{3}+8q^{2}-3q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-2z^{6}a^{-6}+z^{4}a^{-2}-z^{4}a^{-4}-6z^{4}a^{-6}+3z^{4}a^{-8}+2z^{2}a^{-2}+2z^{2}a^{-4}-7z^{2}a^{-6}+5z^{2}a^{-8}-z^{2}a^{-10}+a^{-2}+2a^{-4}-3a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+4z^{9}a^{-5}+9z^{9}a^{-7}+5z^{9}a^{-9}+5z^{8}a^{-4}+16z^{8}a^{-6}+21z^{8}a^{-8}+10z^{8}a^{-10}+3z^{7}a^{-3}+z^{7}a^{-5}+2z^{7}a^{-7}+14z^{7}a^{-9}+10z^{7}a^{-11}+z^{6}a^{-2}-10z^{6}a^{-4}-41z^{6}a^{-6}-44z^{6}a^{-8}-9z^{6}a^{-10}+5z^{6}a^{-12}-7z^{5}a^{-3}-17z^{5}a^{-5}-38z^{5}a^{-7}-44z^{5}a^{-9}-15z^{5}a^{-11}+z^{5}a^{-13}-3z^{4}a^{-2}+7z^{4}a^{-4}+36z^{4}a^{-6}+27z^{4}a^{-8}-4z^{4}a^{-10}-5z^{4}a^{-12}+5z^{3}a^{-3}+18z^{3}a^{-5}+37z^{3}a^{-7}+30z^{3}a^{-9}+6z^{3}a^{-11}+3z^{2}a^{-2}-4z^{2}a^{-4}-15z^{2}a^{-6}-6z^{2}a^{-8}+2z^{2}a^{-10}-za^{-3}-7za^{-5}-10za^{-7}-4za^{-9}-a^{-2}+2a^{-4}+3a^{-6}+a^{-8}$
The A2 invariant	Data:K11a100/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a100/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a100 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["K11a100"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{3}+15t^{2}-32t+41-32t^{-1}+15t^{-2}-3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 141, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{11}+5q^{10}-10q^{9}+15q^{8}-21q^{7}+23q^{6}-22q^{5}+19q^{4}-13q^{3}+8q^{2}-3q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-4}-2z^{6}a^{-6}+z^{4}a^{-2}-z^{4}a^{-4}-6z^{4}a^{-6}+3z^{4}a^{-8}+2z^{2}a^{-2}+2z^{2}a^{-4}-7z^{2}a^{-6}+5z^{2}a^{-8}-z^{2}a^{-10}+a^{-2}+2a^{-4}-3a^{-6}+a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+4z^{9}a^{-5}+9z^{9}a^{-7}+5z^{9}a^{-9}+5z^{8}a^{-4}+16z^{8}a^{-6}+21z^{8}a^{-8}+10z^{8}a^{-10}+3z^{7}a^{-3}+z^{7}a^{-5}+2z^{7}a^{-7}+14z^{7}a^{-9}+10z^{7}a^{-11}+z^{6}a^{-2}-10z^{6}a^{-4}-41z^{6}a^{-6}-44z^{6}a^{-8}-9z^{6}a^{-10}+5z^{6}a^{-12}-7z^{5}a^{-3}-17z^{5}a^{-5}-38z^{5}a^{-7}-44z^{5}a^{-9}-15z^{5}a^{-11}+z^{5}a^{-13}-3z^{4}a^{-2}+7z^{4}a^{-4}+36z^{4}a^{-6}+27z^{4}a^{-8}-4z^{4}a^{-10}-5z^{4}a^{-12}+5z^{3}a^{-3}+18z^{3}a^{-5}+37z^{3}a^{-7}+30z^{3}a^{-9}+6z^{3}a^{-11}+3z^{2}a^{-2}-4z^{2}a^{-4}-15z^{2}a^{-6}-6z^{2}a^{-8}+2z^{2}a^{-10}-za^{-3}-7za^{-5}-10za^{-7}-4za^{-9}-a^{-2}+2a^{-4}+3a^{-6}+a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a290,}

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["K11a100"];

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]=

{ $-3t^{3}+15t^{2}-32t+41-32t^{-1}+15t^{-2}-3t^{-3}$, $-q^{11}+5q^{10}-10q^{9}+15q^{8}-21q^{7}+23q^{6}-22q^{5}+19q^{4}-13q^{3}+8q^{2}-3q+1$ }

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]=

{K11a290,}

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:

(1, 0)

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 $4$ $0$ $8$ ${\frac {14}{3}}$ ${\frac {82}{3}}$ $0$ $64$ $64$ $96$ ${\frac {32}{3}}$ $0$ ${\frac {56}{3}}$ ${\frac {328}{3}}$ ${\frac {9151}{30}}$ $-{\frac {2542}{15}}$ ${\frac {30782}{45}}$ ${\frac {1313}{18}}$ ${\frac {1951}{30}}$

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=$4 is the signature of K11a100. 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 23                       1 -1 21                     4   4 19                   6 1   -5 17                 9 4     5 15               12 6       -6 13             11 9         2 11           11 12           1 9         8 11             -3 7       5 11               6 5     3 8                 -5 3   1 6                   5 1   2                     -2 -1 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a99 K11a101