K11a82

K11a81

K11a83

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a82 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a82/ThurstonBennequinNumber Hyperbolic Volume 13.8883 A-Polynomial See Data:K11a82/A-polynomial

[edit Notes for K11a82's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a82's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 95, -2 }
Jones polynomial	$-q^{4}+3q^{3}-6q^{2}+10q-12+15q^{-1}-15q^{-2}+13q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-14a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+5a^{4}z^{2}-15a^{2}z^{2}-3z^{2}a^{-2}+13z^{2}+2a^{4}-6a^{2}-2a^{-2}+7$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+7az^{9}+3z^{9}a^{-1}+7a^{4}z^{8}+11a^{2}z^{8}+3z^{8}a^{-2}+7z^{8}+7a^{5}z^{7}-15az^{7}-7z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-14a^{4}z^{6}-43a^{2}z^{6}-12z^{6}a^{-2}-36z^{6}+3a^{7}z^{5}-12a^{5}z^{5}-21a^{3}z^{5}-6az^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}+14a^{4}z^{4}+51a^{2}z^{4}+15z^{4}a^{-2}+46z^{4}-3a^{7}z^{3}+11a^{5}z^{3}+28a^{3}z^{3}+22az^{3}+13z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+a^{6}z^{2}-7a^{4}z^{2}-27a^{2}z^{2}-8z^{2}a^{-2}-26z^{2}-4a^{5}z-10a^{3}z-10az-6za^{-1}-2za^{-3}+2a^{4}+6a^{2}+2a^{-2}+7$
The A2 invariant	$q^{20}-q^{18}+2q^{16}-q^{14}-q^{12}+q^{10}-4q^{8}+2q^{6}-2q^{4}+2q^{2}+3+3q^{-4}-q^{-6}-q^{-12}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+5q^{106}-3q^{104}-2q^{102}+10q^{100}-17q^{98}+24q^{96}-27q^{94}+19q^{92}-7q^{90}-13q^{88}+37q^{86}-55q^{84}+66q^{82}-62q^{80}+39q^{78}-q^{76}-44q^{74}+91q^{72}-119q^{70}+122q^{68}-90q^{66}+26q^{64}+56q^{62}-124q^{60}+166q^{58}-152q^{56}+86q^{54}+9q^{52}-103q^{50}+148q^{48}-128q^{46}+50q^{44}+54q^{42}-134q^{40}+144q^{38}-84q^{36}-38q^{34}+161q^{32}-239q^{30}+217q^{28}-113q^{26}-48q^{24}+202q^{22}-291q^{20}+282q^{18}-182q^{16}+24q^{14}+130q^{12}-230q^{10}+246q^{8}-165q^{6}+40q^{4}+89q^{2}-159+157q^{-2}-70q^{-4}-45q^{-6}+148q^{-8}-183q^{-10}+139q^{-12}-24q^{-14}-108q^{-16}+209q^{-18}-228q^{-20}+170q^{-22}-58q^{-24}-69q^{-26}+156q^{-28}-184q^{-30}+152q^{-32}-79q^{-34}-q^{-36}+56q^{-38}-81q^{-40}+72q^{-42}-47q^{-44}+19q^{-46}+3q^{-48}-15q^{-50}+14q^{-52}-11q^{-54}+6q^{-56}-2q^{-58}+q^{-60}$

Further Quantum Invariants

K11a82 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["K11a82"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 95, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+3q^{3}-6q^{2}+10q-12+15q^{-1}-15q^{-2}+13q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+2z^{6}+4a^{4}z^{4}-14a^{2}z^{4}-z^{4}a^{-2}+9z^{4}+5a^{4}z^{2}-15a^{2}z^{2}-3z^{2}a^{-2}+13z^{2}+2a^{4}-6a^{2}-2a^{-2}+7$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+7az^{9}+3z^{9}a^{-1}+7a^{4}z^{8}+11a^{2}z^{8}+3z^{8}a^{-2}+7z^{8}+7a^{5}z^{7}-15az^{7}-7z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-14a^{4}z^{6}-43a^{2}z^{6}-12z^{6}a^{-2}-36z^{6}+3a^{7}z^{5}-12a^{5}z^{5}-21a^{3}z^{5}-6az^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}+14a^{4}z^{4}+51a^{2}z^{4}+15z^{4}a^{-2}+46z^{4}-3a^{7}z^{3}+11a^{5}z^{3}+28a^{3}z^{3}+22az^{3}+13z^{3}a^{-1}+5z^{3}a^{-3}-a^{8}z^{2}+a^{6}z^{2}-7a^{4}z^{2}-27a^{2}z^{2}-8z^{2}a^{-2}-26z^{2}-4a^{5}z-10a^{3}z-10az-6za^{-1}-2za^{-3}+2a^{4}+6a^{2}+2a^{-2}+7$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_116, K11a7, K11a33,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a33,}

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["K11a82"];

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]=

{ $-t^{4}+5t^{3}-12t^{2}+19t-21+19t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$, $-q^{4}+3q^{3}-6q^{2}+10q-12+15q^{-1}-15q^{-2}+13q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$ }

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_116, K11a7, K11a33,}

In[6]:=

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

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

Out[6]=

{K11a33,}

Vassiliev invariants

V2 and V3:

(0, 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 $0$ $16$ $0$ $0$ $16$ $0$ $-{\frac {224}{3}}$ $-{\frac {224}{3}}$ $-16$ $0$ $128$ $0$ $0$ $352$ ${\frac {784}{3}}$ $-{\frac {208}{3}}$ ${\frac {128}{3}}$ $-64$

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 K11a82. 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 9                       1 -1 7                     2   2 5                   4 1   -3 3                 6 2     4 1               6 4       -2 -1             9 6         3 -3           7 7           0 -5         6 8             -2 -7       4 7               3 -9     2 6                 -4 -11   1 4                   3 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a81 K11a83