K11a300

K11a299

K11a301

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a300 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,4,11,3 X18,5,19,6 X14,7,15,8 X2,10,3,9 X20,12,21,11 X22,13,1,14 X8,15,9,16 X12,18,13,17 X4,19,5,20 X16,21,17,22
Gauss code	1, -5, 2, -10, 3, -1, 4, -8, 5, -2, 6, -9, 7, -4, 8, -11, 9, -3, 10, -6, 11, -7
Dowker-Thistlethwaite code	6 10 18 14 2 20 22 8 12 4 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a300/ThurstonBennequinNumber Hyperbolic Volume 17.9979 A-Polynomial See Data:K11a300/A-polynomial

[edit Notes for K11a300's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 0

[edit Notes for K11a300's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+17t^{2}-32t+41-32t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 153, 0 }
Jones polynomial	$-q^{5}+4q^{4}-9q^{3}+16q^{2}-21q+25-25q^{-1}+21q^{-2}-16q^{-3}+10q^{-4}-4q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-7a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+2a^{4}z^{2}-9a^{2}z^{2}-3z^{2}a^{-2}+8z^{2}+2a^{4}-4a^{2}+3$
Kauffman polynomial (db, data sources)	$3a^{2}z^{10}+3z^{10}+8a^{3}z^{9}+17az^{9}+9z^{9}a^{-1}+8a^{4}z^{8}+13a^{2}z^{8}+11z^{8}a^{-2}+16z^{8}+4a^{5}z^{7}-13a^{3}z^{7}-35az^{7}-10z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-18a^{4}z^{6}-49a^{2}z^{6}-19z^{6}a^{-2}+4z^{6}a^{-4}-53z^{6}-8a^{5}z^{5}+a^{3}z^{5}+20az^{5}-z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+13a^{4}z^{4}+52a^{2}z^{4}+15z^{4}a^{-2}-5z^{4}a^{-4}+57z^{4}+4a^{5}z^{3}+2a^{3}z^{3}+7z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-8a^{4}z^{2}-25a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-23z^{2}-a^{5}z-2za^{-1}-za^{-3}+2a^{4}+4a^{2}+3$
The A2 invariant	$q^{18}-q^{16}+q^{14}+3q^{12}-4q^{10}+3q^{8}-3q^{6}-2q^{4}+3q^{2}-4+6q^{-2}-3q^{-4}+2q^{-6}+3q^{-8}-3q^{-10}+2q^{-12}-q^{-14}$
The G2 invariant	Data:K11a300/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a300 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["K11a300"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+17t^{2}-32t+41-32t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+2z^{6}+z^{4}-2z^{2}+1$

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]=

{ 153, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-9q^{3}+16q^{2}-21q+25-25q^{-1}+21q^{-2}-16q^{-3}+10q^{-4}-4q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-7a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+2a^{4}z^{2}-9a^{2}z^{2}-3z^{2}a^{-2}+8z^{2}+2a^{4}-4a^{2}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{2}z^{10}+3z^{10}+8a^{3}z^{9}+17az^{9}+9z^{9}a^{-1}+8a^{4}z^{8}+13a^{2}z^{8}+11z^{8}a^{-2}+16z^{8}+4a^{5}z^{7}-13a^{3}z^{7}-35az^{7}-10z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-18a^{4}z^{6}-49a^{2}z^{6}-19z^{6}a^{-2}+4z^{6}a^{-4}-53z^{6}-8a^{5}z^{5}+a^{3}z^{5}+20az^{5}-z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+13a^{4}z^{4}+52a^{2}z^{4}+15z^{4}a^{-2}-5z^{4}a^{-4}+57z^{4}+4a^{5}z^{3}+2a^{3}z^{3}+7z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-8a^{4}z^{2}-25a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-23z^{2}-a^{5}z-2za^{-1}-za^{-3}+2a^{4}+4a^{2}+3$

"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["K11a300"];

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]=

{ $t^{4}-6t^{3}+17t^{2}-32t+41-32t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$, $-q^{5}+4q^{4}-9q^{3}+16q^{2}-21q+25-25q^{-1}+21q^{-2}-16q^{-3}+10q^{-4}-4q^{-5}+q^{-6}$ }

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, 2)

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 $-8$ $16$ $32$ ${\frac {68}{3}}$ ${\frac {28}{3}}$ $-128$ $-{\frac {704}{3}}$ $-{\frac {224}{3}}$ $-48$ $-{\frac {256}{3}}$ $128$ $-{\frac {544}{3}}$ $-{\frac {224}{3}}$ ${\frac {3089}{15}}$ $-{\frac {1396}{15}}$ ${\frac {9476}{45}}$ $-{\frac {65}{9}}$ ${\frac {449}{15}}$

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=$0 is the signature of K11a300. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 10 3     7 3               11 6       -5 1             14 10         4 -1           12 12           0 -3         9 13             -4 -5       7 12               5 -7     3 9                 -6 -9   1 7                   6 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a299 K11a301