K11a176

K11a175

K11a177

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a176 at Knotilus!

Knot presentations

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

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:K11a176/ThurstonBennequinNumber Hyperbolic Volume 14.1348 A-Polynomial See Data:K11a176/A-polynomial

[edit Notes for K11a176'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 K11a176's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-13t^{2}+23t-27+23t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 111, 2 }
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+12q^{5}-16q^{4}+18q^{3}-17q^{2}+15q-10+6q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-6z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-14z^{4}a^{-2}+8z^{4}a^{-4}-z^{4}a^{-6}+4z^{4}-13z^{2}a^{-2}+10z^{2}a^{-4}-2z^{2}a^{-6}+5z^{2}-3a^{-2}+3a^{-4}-a^{-6}+2$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+7z^{9}a^{-3}+4z^{9}a^{-5}+8z^{8}a^{-2}+11z^{8}a^{-4}+7z^{8}a^{-6}+4z^{8}+3az^{7}-2z^{7}a^{-1}-10z^{7}a^{-3}+2z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-26z^{6}a^{-2}-29z^{6}a^{-4}-10z^{6}a^{-6}+4z^{6}a^{-8}-10z^{6}-9az^{5}-7z^{5}a^{-1}+z^{5}a^{-3}-14z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+32z^{4}a^{-2}+31z^{4}a^{-4}+4z^{4}a^{-6}-6z^{4}a^{-8}+8z^{4}+7az^{3}+7z^{3}a^{-1}+6z^{3}a^{-3}+12z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}+2a^{2}z^{2}-19z^{2}a^{-2}-15z^{2}a^{-4}-2z^{2}a^{-6}+z^{2}a^{-8}-5z^{2}-az-2za^{-1}-2za^{-3}-2za^{-5}-za^{-7}+3a^{-2}+3a^{-4}+a^{-6}+2$
The A2 invariant	$q^{8}-q^{6}+2q^{4}-q^{2}-1+3q^{-2}-3q^{-4}+4q^{-6}-q^{-8}+q^{-12}-3q^{-14}+3q^{-16}-q^{-18}+q^{-22}-q^{-24}$
The G2 invariant	$q^{46}-2q^{44}+5q^{42}-9q^{40}+10q^{38}-10q^{36}+2q^{34}+15q^{32}-33q^{30}+54q^{28}-64q^{26}+54q^{24}-22q^{22}-37q^{20}+107q^{18}-165q^{16}+190q^{14}-155q^{12}+60q^{10}+81q^{8}-220q^{6}+320q^{4}-326q^{2}+227-48q^{-2}-161q^{-4}+313q^{-6}-357q^{-8}+279q^{-10}-98q^{-12}-98q^{-14}+232q^{-16}-254q^{-18}+142q^{-20}+45q^{-22}-225q^{-24}+306q^{-26}-245q^{-28}+52q^{-30}+202q^{-32}-413q^{-34}+503q^{-36}-419q^{-38}+184q^{-40}+124q^{-42}-401q^{-44}+544q^{-46}-504q^{-48}+309q^{-50}-29q^{-52}-222q^{-54}+359q^{-56}-340q^{-58}+191q^{-60}+17q^{-62}-189q^{-64}+243q^{-66}-167q^{-68}-6q^{-70}+197q^{-72}-313q^{-74}+315q^{-76}-192q^{-78}-7q^{-80}+203q^{-82}-339q^{-84}+362q^{-86}-274q^{-88}+118q^{-90}+50q^{-92}-179q^{-94}+237q^{-96}-219q^{-98}+152q^{-100}-60q^{-102}-23q^{-104}+72q^{-106}-92q^{-108}+77q^{-110}-48q^{-112}+22q^{-114}+2q^{-116}-13q^{-118}+15q^{-120}-13q^{-122}+7q^{-124}-3q^{-126}+q^{-128}$

Further Quantum Invariants

K11a176 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["K11a176"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-13t^{2}+23t-27+23t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 111, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-8q^{6}+12q^{5}-16q^{4}+18q^{3}-17q^{2}+15q-10+6q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-6z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-14z^{4}a^{-2}+8z^{4}a^{-4}-z^{4}a^{-6}+4z^{4}-13z^{2}a^{-2}+10z^{2}a^{-4}-2z^{2}a^{-6}+5z^{2}-3a^{-2}+3a^{-4}-a^{-6}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+7z^{9}a^{-3}+4z^{9}a^{-5}+8z^{8}a^{-2}+11z^{8}a^{-4}+7z^{8}a^{-6}+4z^{8}+3az^{7}-2z^{7}a^{-1}-10z^{7}a^{-3}+2z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-26z^{6}a^{-2}-29z^{6}a^{-4}-10z^{6}a^{-6}+4z^{6}a^{-8}-10z^{6}-9az^{5}-7z^{5}a^{-1}+z^{5}a^{-3}-14z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+32z^{4}a^{-2}+31z^{4}a^{-4}+4z^{4}a^{-6}-6z^{4}a^{-8}+8z^{4}+7az^{3}+7z^{3}a^{-1}+6z^{3}a^{-3}+12z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}+2a^{2}z^{2}-19z^{2}a^{-2}-15z^{2}a^{-4}-2z^{2}a^{-6}+z^{2}a^{-8}-5z^{2}-az-2za^{-1}-2za^{-3}-2za^{-5}-za^{-7}+3a^{-2}+3a^{-4}+a^{-6}+2$

"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["K11a176"];

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]=

{ $-t^{4}+5t^{3}-13t^{2}+23t-27+23t^{-1}-13t^{-2}+5t^{-3}-t^{-4}$, $-q^{8}+4q^{7}-8q^{6}+12q^{5}-16q^{4}+18q^{3}-17q^{2}+15q-10+6q^{-1}-3q^{-2}+q^{-3}$ }

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:

(0, 1)

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 $0$ $8$ $0$ $48$ $24$ $0$ ${\frac {272}{3}}$ $-{\frac {64}{3}}$ $72$ $0$ $32$ $0$ $0$ $168$ $-24$ $40$ $72$ $-40$

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 K11a176. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a175 K11a177