K11a206

K11a205

K11a207

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a206 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a206/ThurstonBennequinNumber Hyperbolic Volume 8.62876 A-Polynomial See Data:K11a206/A-polynomial

[edit Notes for K11a206's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 6

[edit Notes for K11a206's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-7t^{2}+7t-7+7t^{-1}-7t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}+3z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 47, -6 }
Jones polynomial	$q^{-1}-2q^{-2}+3q^{-3}-4q^{-4}+6q^{-5}-6q^{-6}+7q^{-7}-6q^{-8}+5q^{-9}-4q^{-10}+2q^{-11}-q^{-12}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{10}-4z^{2}a^{10}-3a^{10}+2z^{6}a^{8}+10z^{4}a^{8}+13z^{2}a^{8}+4a^{8}-z^{8}a^{6}-6z^{6}a^{6}-11z^{4}a^{6}-7z^{2}a^{6}-a^{6}+z^{6}a^{4}+5z^{4}a^{4}+6z^{2}a^{4}+a^{4}$
Kauffman polynomial (db, data sources)	$z^{3}a^{15}-za^{15}+2z^{4}a^{14}-z^{2}a^{14}+3z^{5}a^{13}-3z^{3}a^{13}+2za^{13}+3z^{6}a^{12}-3z^{4}a^{12}+z^{2}a^{12}+3z^{7}a^{11}-5z^{5}a^{11}+2z^{3}a^{11}-za^{11}+3z^{8}a^{10}-9z^{6}a^{10}+11z^{4}a^{10}-11z^{2}a^{10}+3a^{10}+2z^{9}a^{9}-5z^{7}a^{9}-z^{5}a^{9}+5z^{3}a^{9}-3za^{9}+z^{10}a^{8}-z^{8}a^{8}-10z^{6}a^{8}+19z^{4}a^{8}-13z^{2}a^{8}+4a^{8}+4z^{9}a^{7}-20z^{7}a^{7}+29z^{5}a^{7}-14z^{3}a^{7}+2za^{7}+z^{10}a^{6}-3z^{8}a^{6}-4z^{6}a^{6}+14z^{4}a^{6}-7z^{2}a^{6}+a^{6}+2z^{9}a^{5}-12z^{7}a^{5}+22z^{5}a^{5}-13z^{3}a^{5}+za^{5}+z^{8}a^{4}-6z^{6}a^{4}+11z^{4}a^{4}-7z^{2}a^{4}+a^{4}$
The A2 invariant	$-q^{36}-q^{34}-q^{30}-q^{26}+q^{24}+q^{22}+q^{20}+3q^{18}+q^{14}-q^{12}+q^{4}$
The G2 invariant	$q^{196}-q^{194}+2q^{192}-2q^{190}+q^{188}-2q^{184}+5q^{182}-6q^{180}+6q^{178}-5q^{176}+q^{174}+3q^{172}-7q^{170}+11q^{168}-10q^{166}+7q^{164}-3q^{162}-3q^{160}+6q^{158}-9q^{156}+9q^{154}-8q^{152}+3q^{150}+q^{148}-6q^{146}+6q^{144}-5q^{142}+2q^{140}-2q^{138}-q^{136}-q^{134}-q^{130}+q^{126}-4q^{124}+3q^{122}-4q^{120}+q^{118}-3q^{112}+6q^{110}-4q^{108}-2q^{106}+9q^{104}-14q^{102}+16q^{100}-9q^{98}+3q^{96}+8q^{94}-11q^{92}+19q^{90}-13q^{88}+9q^{86}-q^{84}-3q^{82}+9q^{80}-7q^{78}+7q^{76}-q^{74}-2q^{72}+4q^{70}-5q^{68}-q^{66}+8q^{64}-14q^{62}+13q^{60}-8q^{58}-2q^{56}+13q^{54}-20q^{52}+19q^{50}-14q^{48}+4q^{46}+6q^{44}-13q^{42}+15q^{40}-10q^{38}+7q^{36}-2q^{32}+3q^{30}-4q^{28}+3q^{26}-q^{24}+q^{22}$

Further Quantum Invariants

K11a206 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["K11a206"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-7t^{2}+7t-7+7t^{-1}-7t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}+3z^{4}+8z^{2}+1$

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]=

{ 47, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-1}-2q^{-2}+3q^{-3}-4q^{-4}+6q^{-5}-6q^{-6}+7q^{-7}-6q^{-8}+5q^{-9}-4q^{-10}+2q^{-11}-q^{-12}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{10}-4z^{2}a^{10}-3a^{10}+2z^{6}a^{8}+10z^{4}a^{8}+13z^{2}a^{8}+4a^{8}-z^{8}a^{6}-6z^{6}a^{6}-11z^{4}a^{6}-7z^{2}a^{6}-a^{6}+z^{6}a^{4}+5z^{4}a^{4}+6z^{2}a^{4}+a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{15}-za^{15}+2z^{4}a^{14}-z^{2}a^{14}+3z^{5}a^{13}-3z^{3}a^{13}+2za^{13}+3z^{6}a^{12}-3z^{4}a^{12}+z^{2}a^{12}+3z^{7}a^{11}-5z^{5}a^{11}+2z^{3}a^{11}-za^{11}+3z^{8}a^{10}-9z^{6}a^{10}+11z^{4}a^{10}-11z^{2}a^{10}+3a^{10}+2z^{9}a^{9}-5z^{7}a^{9}-z^{5}a^{9}+5z^{3}a^{9}-3za^{9}+z^{10}a^{8}-z^{8}a^{8}-10z^{6}a^{8}+19z^{4}a^{8}-13z^{2}a^{8}+4a^{8}+4z^{9}a^{7}-20z^{7}a^{7}+29z^{5}a^{7}-14z^{3}a^{7}+2za^{7}+z^{10}a^{6}-3z^{8}a^{6}-4z^{6}a^{6}+14z^{4}a^{6}-7z^{2}a^{6}+a^{6}+2z^{9}a^{5}-12z^{7}a^{5}+22z^{5}a^{5}-13z^{3}a^{5}+za^{5}+z^{8}a^{4}-6z^{6}a^{4}+11z^{4}a^{4}-7z^{2}a^{4}+a^{4}$

"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["K11a206"];

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]=

{ $-t^{4}+5t^{3}-7t^{2}+7t-7+7t^{-1}-7t^{-2}+5t^{-3}-t^{-4}$, $q^{-1}-2q^{-2}+3q^{-3}-4q^{-4}+6q^{-5}-6q^{-6}+7q^{-7}-6q^{-8}+5q^{-9}-4q^{-10}+2q^{-11}-q^{-12}$ }

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:

(8, -23)

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 $32$ $-184$ $512$ ${\frac {4048}{3}}$ ${\frac {680}{3}}$ $-5888$ $-{\frac {32656}{3}}$ $-{\frac {5728}{3}}$ $-1624$ ${\frac {16384}{3}}$ $16928$ ${\frac {129536}{3}}$ ${\frac {21760}{3}}$ ${\frac {1334044}{15}}$ ${\frac {6184}{15}}$ ${\frac {1694776}{45}}$ ${\frac {6308}{9}}$ ${\frac {77164}{15}}$

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 $s=$-6 is the signature of K11a206. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ -1                       1 1 -3                     1   -1 -5                   2 1   1 -7                 3 2     -1 -9               3 1       2 -11             3 3         0 -13           4 3           1 -15         2 3             1 -17       3 4               -1 -19     1 2                 1 -21   1 3                   -2 -23   1                     1 -25 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a205 K11a207