K11a41

K11a40

K11a42

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a41 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X18,10,19,9 X20,12,21,11 X16,13,17,14 X6,15,7,16 X22,18,1,17 X12,20,13,19 X10,22,11,21
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -7, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code	4 8 14 2 18 20 16 6 22 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a41/ThurstonBennequinNumber Hyperbolic Volume 14.6854 A-Polynomial See Data:K11a41/A-polynomial

[edit Notes for K11a41'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 K11a41's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-11t^{2}+27t-35+27t^{-1}-11t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 115, 2 }
Jones polynomial	$q^{9}-3q^{8}+6q^{7}-11q^{6}+15q^{5}-18q^{4}+19q^{3}-16q^{2}+13q-8+4q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}a^{-6}-z^{4}+2z^{2}a^{-2}+3z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}-z^{2}+a^{-2}+2a^{-4}-3a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-4}+z^{10}a^{-6}+4z^{9}a^{-3}+7z^{9}a^{-5}+3z^{9}a^{-7}+7z^{8}a^{-2}+12z^{8}a^{-4}+9z^{8}a^{-6}+4z^{8}a^{-8}+7z^{7}a^{-1}+4z^{7}a^{-3}-6z^{7}a^{-5}+3z^{7}a^{-9}-8z^{6}a^{-2}-29z^{6}a^{-4}-27z^{6}a^{-6}-9z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-11z^{5}a^{-1}-18z^{5}a^{-3}-12z^{5}a^{-5}-15z^{5}a^{-7}-9z^{5}a^{-9}+26z^{4}a^{-4}+28z^{4}a^{-6}+5z^{4}a^{-8}-3z^{4}a^{-10}-6z^{4}-az^{3}+4z^{3}a^{-1}+14z^{3}a^{-3}+20z^{3}a^{-5}+19z^{3}a^{-7}+8z^{3}a^{-9}+z^{2}a^{-2}-10z^{2}a^{-4}-13z^{2}a^{-6}-2z^{2}a^{-8}+2z^{2}a^{-10}+2z^{2}-za^{-1}-3za^{-3}-7za^{-5}-8za^{-7}-3za^{-9}-a^{-2}+2a^{-4}+3a^{-6}+a^{-8}$
The A2 invariant	Data:K11a41/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a41/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a41 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["K11a41"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-11t^{2}+27t-35+27t^{-1}-11t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 115, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{9}-3q^{8}+6q^{7}-11q^{6}+15q^{5}-18q^{4}+19q^{3}-16q^{2}+13q-8+4q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-4}+z^{10}a^{-6}+4z^{9}a^{-3}+7z^{9}a^{-5}+3z^{9}a^{-7}+7z^{8}a^{-2}+12z^{8}a^{-4}+9z^{8}a^{-6}+4z^{8}a^{-8}+7z^{7}a^{-1}+4z^{7}a^{-3}-6z^{7}a^{-5}+3z^{7}a^{-9}-8z^{6}a^{-2}-29z^{6}a^{-4}-27z^{6}a^{-6}-9z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-11z^{5}a^{-1}-18z^{5}a^{-3}-12z^{5}a^{-5}-15z^{5}a^{-7}-9z^{5}a^{-9}+26z^{4}a^{-4}+28z^{4}a^{-6}+5z^{4}a^{-8}-3z^{4}a^{-10}-6z^{4}-az^{3}+4z^{3}a^{-1}+14z^{3}a^{-3}+20z^{3}a^{-5}+19z^{3}a^{-7}+8z^{3}a^{-9}+z^{2}a^{-2}-10z^{2}a^{-4}-13z^{2}a^{-6}-2z^{2}a^{-8}+2z^{2}a^{-10}+2z^{2}-za^{-1}-3za^{-3}-7za^{-5}-8za^{-7}-3za^{-9}-a^{-2}+2a^{-4}+3a^{-6}+a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_121, K11a183, K11a198, K11a331,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a183,}

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["K11a41"];

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]=

{ $2t^{3}-11t^{2}+27t-35+27t^{-1}-11t^{-2}+2t^{-3}$, $q^{9}-3q^{8}+6q^{7}-11q^{6}+15q^{5}-18q^{4}+19q^{3}-16q^{2}+13q-8+4q^{-1}-q^{-2}$ }

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]=

{10_121, K11a183, K11a198, K11a331,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a183,}

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 $4$ $0$ $8$ $-{\frac {82}{3}}$ $-{\frac {14}{3}}$ $0$ $-96$ $0$ $0$ ${\frac {32}{3}}$ $0$ $-{\frac {328}{3}}$ $-{\frac {56}{3}}$ $-{\frac {7649}{30}}$ $-{\frac {714}{5}}$ ${\frac {8222}{45}}$ $-{\frac {223}{18}}$ ${\frac {991}{30}}$

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=$2 is the signature of K11a41. 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                     2   -2 15                   4 1   3 13                 7 2     -5 11               8 4       4 9             10 7         -3 7           9 8           1 5         7 10             3 3       6 9               -3 1     3 8                 5 -1   1 5                   -4 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a40 K11a42