K11a180

K11a179

K11a181

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a180 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a180/ThurstonBennequinNumber Hyperbolic Volume 12.7428 A-Polynomial See Data:K11a180/A-polynomial

[edit Notes for K11a180'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 K11a180's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+11t^{2}-17t+21-17t^{-1}+11t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 89, 0 }
Jones polynomial	$-q^{5}+3q^{4}-5q^{3}+9q^{2}-12q+14-14q^{-1}+12q^{-2}-9q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+6z^{6}+a^{4}z^{4}-9a^{2}z^{4}-4z^{4}a^{-2}+13z^{4}+3a^{4}z^{2}-12a^{2}z^{2}-4z^{2}a^{-2}+11z^{2}+2a^{4}-4a^{2}+3$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+6az^{9}+3z^{9}a^{-1}+4a^{4}z^{8}+5a^{2}z^{8}+4z^{8}a^{-2}+5z^{8}+3a^{5}z^{7}-5a^{3}z^{7}-16az^{7}-4z^{7}a^{-1}+4z^{7}a^{-3}+a^{6}z^{6}-11a^{4}z^{6}-22a^{2}z^{6}-7z^{6}a^{-2}+3z^{6}a^{-4}-20z^{6}-9a^{5}z^{5}-a^{3}z^{5}+18az^{5}+z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+8a^{4}z^{4}+31a^{2}z^{4}+4z^{4}a^{-2}-7z^{4}a^{-4}+31z^{4}+6a^{5}z^{3}+a^{3}z^{3}-6az^{3}+5z^{3}a^{-1}+4z^{3}a^{-3}-2z^{3}a^{-5}+2a^{6}z^{2}-5a^{4}z^{2}-20a^{2}z^{2}+3z^{2}a^{-4}-16z^{2}-a^{5}z-2za^{-1}-za^{-3}+2a^{4}+4a^{2}+3$
The A2 invariant	$q^{18}+q^{12}-2q^{10}+2q^{8}-q^{6}-q^{4}+q^{2}-3+3q^{-2}-q^{-4}+2q^{-6}+2q^{-8}-q^{-10}+q^{-12}-q^{-14}$
The G2 invariant	$q^{94}-2q^{92}+5q^{90}-9q^{88}+10q^{86}-9q^{84}+16q^{80}-32q^{78}+48q^{76}-53q^{74}+38q^{72}-5q^{70}-43q^{68}+94q^{66}-122q^{64}+121q^{62}-76q^{60}-2q^{58}+89q^{56}-155q^{54}+177q^{52}-139q^{50}+54q^{48}+47q^{46}-126q^{44}+151q^{42}-112q^{40}+33q^{38}+56q^{36}-112q^{34}+105q^{32}-46q^{30}-52q^{28}+139q^{26}-177q^{24}+144q^{22}-49q^{20}-78q^{18}+186q^{16}-241q^{14}+217q^{12}-125q^{10}-9q^{8}+131q^{6}-206q^{4}+206q^{2}-135+32q^{-2}+67q^{-4}-120q^{-6}+113q^{-8}-52q^{-10}-25q^{-12}+91q^{-14}-105q^{-16}+68q^{-18}+8q^{-20}-85q^{-22}+140q^{-24}-141q^{-26}+99q^{-28}-27q^{-30}-53q^{-32}+108q^{-34}-132q^{-36}+119q^{-38}-74q^{-40}+21q^{-42}+27q^{-44}-62q^{-46}+75q^{-48}-70q^{-50}+50q^{-52}-23q^{-54}-3q^{-56}+22q^{-58}-31q^{-60}+30q^{-62}-21q^{-64}+13q^{-66}-2q^{-68}-5q^{-70}+6q^{-72}-7q^{-74}+4q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

K11a180 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["K11a180"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+11t^{2}-17t+21-17t^{-1}+11t^{-2}-5t^{-3}+t^{-4}$

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]=

{ 89, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-5q^{3}+9q^{2}-12q+14-14q^{-1}+12q^{-2}-9q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+6z^{6}+a^{4}z^{4}-9a^{2}z^{4}-4z^{4}a^{-2}+13z^{4}+3a^{4}z^{2}-12a^{2}z^{2}-4z^{2}a^{-2}+11z^{2}+2a^{4}-4a^{2}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+6az^{9}+3z^{9}a^{-1}+4a^{4}z^{8}+5a^{2}z^{8}+4z^{8}a^{-2}+5z^{8}+3a^{5}z^{7}-5a^{3}z^{7}-16az^{7}-4z^{7}a^{-1}+4z^{7}a^{-3}+a^{6}z^{6}-11a^{4}z^{6}-22a^{2}z^{6}-7z^{6}a^{-2}+3z^{6}a^{-4}-20z^{6}-9a^{5}z^{5}-a^{3}z^{5}+18az^{5}+z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+8a^{4}z^{4}+31a^{2}z^{4}+4z^{4}a^{-2}-7z^{4}a^{-4}+31z^{4}+6a^{5}z^{3}+a^{3}z^{3}-6az^{3}+5z^{3}a^{-1}+4z^{3}a^{-3}-2z^{3}a^{-5}+2a^{6}z^{2}-5a^{4}z^{2}-20a^{2}z^{2}+3z^{2}a^{-4}-16z^{2}-a^{5}z-2za^{-1}-za^{-3}+2a^{4}+4a^{2}+3$

"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["K11a180"];

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]=

{ $t^{4}-5t^{3}+11t^{2}-17t+21-17t^{-1}+11t^{-2}-5t^{-3}+t^{-4}$, $-q^{5}+3q^{4}-5q^{3}+9q^{2}-12q+14-14q^{-1}+12q^{-2}-9q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$ }

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 $-8$ $16$ $32$ ${\frac {68}{3}}$ ${\frac {28}{3}}$ $-128$ $-{\frac {704}{3}}$ $-{\frac {320}{3}}$ $-16$ $-{\frac {256}{3}}$ $128$ $-{\frac {544}{3}}$ $-{\frac {224}{3}}$ ${\frac {3089}{15}}$ $-{\frac {2356}{15}}$ ${\frac {12356}{45}}$ $-{\frac {65}{9}}$ ${\frac {929}{15}}$

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 K11a180. 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                     2   2 7                   3 1   -2 5                 6 2     4 3               6 3       -3 1             8 6         2 -1           7 7           0 -3         5 7             -2 -5       4 7               3 -7     2 5                 -3 -9   1 4                   3 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a179 K11a181