K11a182

K11a181

K11a183

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a182 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a182/ThurstonBennequinNumber Hyperbolic Volume 11.2693 A-Polynomial See Data:K11a182/A-polynomial

[edit Notes for K11a182's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a182's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+11t^{2}-13t+13-13t^{-1}+11t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 73, -4 }
Jones polynomial	$-q+3-4q^{-1}+7q^{-2}-9q^{-3}+11q^{-4}-11q^{-5}+10q^{-6}-8q^{-7}+5q^{-8}-3q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{8}+3z^{2}a^{8}+a^{8}-2z^{6}a^{6}-9z^{4}a^{6}-11z^{2}a^{6}-4a^{6}+z^{8}a^{4}+6z^{6}a^{4}+13z^{4}a^{4}+13z^{2}a^{4}+4a^{4}-z^{6}a^{2}-4z^{4}a^{2}-3z^{2}a^{2}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-4z^{3}a^{11}+za^{11}+4z^{6}a^{10}-4z^{4}a^{10}+4z^{7}a^{9}-3z^{5}a^{9}-2z^{3}a^{9}+2za^{9}+4z^{8}a^{8}-7z^{6}a^{8}+7z^{4}a^{8}-4z^{2}a^{8}+a^{8}+3z^{9}a^{7}-6z^{7}a^{7}+5z^{5}a^{7}-2z^{3}a^{7}+z^{10}a^{6}+4z^{8}a^{6}-24z^{6}a^{6}+36z^{4}a^{6}-21z^{2}a^{6}+4a^{6}+6z^{9}a^{5}-23z^{7}a^{5}+27z^{5}a^{5}-11z^{3}a^{5}+z^{10}a^{4}+3z^{8}a^{4}-27z^{6}a^{4}+42z^{4}a^{4}-23z^{2}a^{4}+4a^{4}+3z^{9}a^{3}-12z^{7}a^{3}+12z^{5}a^{3}-4z^{3}a^{3}+za^{3}+3z^{8}a^{2}-14z^{6}a^{2}+18z^{4}a^{2}-7z^{2}a^{2}+z^{7}a-4z^{5}a+3z^{3}a$
The A2 invariant	$q^{30}-q^{26}-2q^{22}+q^{20}-q^{18}+q^{14}-2q^{12}+3q^{10}+2q^{6}+q^{4}+1-q^{-2}$
The G2 invariant	$q^{162}-2q^{160}+4q^{158}-6q^{156}+4q^{154}-2q^{152}-4q^{150}+12q^{148}-18q^{146}+22q^{144}-19q^{142}+8q^{140}+7q^{138}-23q^{136}+37q^{134}-40q^{132}+34q^{130}-20q^{128}-2q^{126}+25q^{124}-39q^{122}+48q^{120}-41q^{118}+29q^{116}-11q^{114}-11q^{112}+28q^{110}-40q^{108}+40q^{106}-30q^{104}+11q^{102}+8q^{100}-24q^{98}+27q^{96}-18q^{94}-2q^{92}+21q^{90}-36q^{88}+25q^{86}+q^{84}-35q^{82}+64q^{80}-71q^{78}+51q^{76}-12q^{74}-39q^{72}+78q^{70}-98q^{68}+84q^{66}-43q^{64}-8q^{62}+55q^{60}-76q^{58}+75q^{56}-48q^{54}+8q^{52}+27q^{50}-49q^{48}+44q^{46}-13q^{44}-19q^{42}+51q^{40}-54q^{38}+32q^{36}+8q^{34}-49q^{32}+79q^{30}-81q^{28}+56q^{26}-9q^{24}-36q^{22}+70q^{20}-76q^{18}+61q^{16}-29q^{14}-3q^{12}+26q^{10}-37q^{8}+35q^{6}-23q^{4}+11q^{2}+1-7q^{-2}+6q^{-4}-7q^{-6}+4q^{-8}-2q^{-10}+q^{-12}$

Further Quantum Invariants

K11a182 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["K11a182"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+11t^{2}-13t+13-13t^{-1}+11t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 73, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{8}+3z^{2}a^{8}+a^{8}-2z^{6}a^{6}-9z^{4}a^{6}-11z^{2}a^{6}-4a^{6}+z^{8}a^{4}+6z^{6}a^{4}+13z^{4}a^{4}+13z^{2}a^{4}+4a^{4}-z^{6}a^{2}-4z^{4}a^{2}-3z^{2}a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-4z^{3}a^{11}+za^{11}+4z^{6}a^{10}-4z^{4}a^{10}+4z^{7}a^{9}-3z^{5}a^{9}-2z^{3}a^{9}+2za^{9}+4z^{8}a^{8}-7z^{6}a^{8}+7z^{4}a^{8}-4z^{2}a^{8}+a^{8}+3z^{9}a^{7}-6z^{7}a^{7}+5z^{5}a^{7}-2z^{3}a^{7}+z^{10}a^{6}+4z^{8}a^{6}-24z^{6}a^{6}+36z^{4}a^{6}-21z^{2}a^{6}+4a^{6}+6z^{9}a^{5}-23z^{7}a^{5}+27z^{5}a^{5}-11z^{3}a^{5}+z^{10}a^{4}+3z^{8}a^{4}-27z^{6}a^{4}+42z^{4}a^{4}-23z^{2}a^{4}+4a^{4}+3z^{9}a^{3}-12z^{7}a^{3}+12z^{5}a^{3}-4z^{3}a^{3}+za^{3}+3z^{8}a^{2}-14z^{6}a^{2}+18z^{4}a^{2}-7z^{2}a^{2}+z^{7}a-4z^{5}a+3z^{3}a$

"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["K11a182"];

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]=

{ $t^{4}-5t^{3}+11t^{2}-13t+13-13t^{-1}+11t^{-2}-5t^{-3}+t^{-4}$, $-q+3-4q^{-1}+7q^{-2}-9q^{-3}+11q^{-4}-11q^{-5}+10q^{-6}-8q^{-7}+5q^{-8}-3q^{-9}+q^{-10}$ }

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 $8$ $-16$ $32$ ${\frac {124}{3}}$ ${\frac {20}{3}}$ $-128$ $-{\frac {448}{3}}$ $-{\frac {256}{3}}$ $16$ ${\frac {256}{3}}$ $128$ ${\frac {992}{3}}$ ${\frac {160}{3}}$ ${\frac {8431}{15}}$ ${\frac {3236}{15}}$ ${\frac {10444}{45}}$ $-{\frac {703}{9}}$ ${\frac {511}{15}}$

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 $s=$-4 is the signature of K11a182. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a181 K11a183