K11a184

K11a183

K11a185

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a184 at Knotilus!

Knot presentations

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

A Braid Representative
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:K11a184/ThurstonBennequinNumber Hyperbolic Volume 12.4499 A-Polynomial See Data:K11a184/A-polynomial

[edit Notes for K11a184'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 K11a184's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-11t^{2}+17t-19+17t^{-1}-11t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 87, -2 }
Jones polynomial	$q^{3}-3q^{2}+5q-8+12q^{-1}-13q^{-2}+14q^{-3}-12q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+9a^{4}z^{4}-13a^{2}z^{4}+4z^{4}-3a^{6}z^{2}+12a^{4}z^{2}-11a^{2}z^{2}+4z^{2}-2a^{6}+4a^{4}-2a^{2}+1$
Kauffman polynomial (db, data sources)	$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+6a^{3}z^{9}+3az^{9}+4a^{6}z^{8}+5a^{4}z^{8}+5a^{2}z^{8}+4z^{8}+4a^{7}z^{7}-4a^{5}z^{7}-17a^{3}z^{7}-6az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-6a^{6}z^{6}-21a^{4}z^{6}-26a^{2}z^{6}+z^{6}a^{-2}-13z^{6}+a^{9}z^{5}-6a^{7}z^{5}+3a^{5}z^{5}+18a^{3}z^{5}-2az^{5}-10z^{5}a^{-1}-6a^{8}z^{4}+4a^{6}z^{4}+34a^{4}z^{4}+39a^{2}z^{4}-3z^{4}a^{-2}+12z^{4}-2a^{9}z^{3}-2a^{3}z^{3}+7az^{3}+7z^{3}a^{-1}+2a^{8}z^{2}-5a^{6}z^{2}-20a^{4}z^{2}-20a^{2}z^{2}+z^{2}a^{-2}-6z^{2}+a^{9}z-a^{5}z-a^{3}z-2az-za^{-1}+2a^{6}+4a^{4}+2a^{2}+1$
The A2 invariant	$-q^{24}-q^{18}+2q^{16}-2q^{14}+q^{12}+q^{10}+4q^{6}-2q^{4}+2q^{2}-1-q^{-2}+q^{-4}-q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+8q^{120}-5q^{118}-3q^{116}+16q^{114}-27q^{112}+36q^{110}-37q^{108}+21q^{106}+2q^{104}-35q^{102}+65q^{100}-82q^{98}+80q^{96}-56q^{94}+11q^{92}+43q^{90}-94q^{88}+126q^{86}-123q^{84}+82q^{82}-19q^{80}-56q^{78}+110q^{76}-126q^{74}+101q^{72}-35q^{70}-37q^{68}+85q^{66}-88q^{64}+40q^{62}+34q^{60}-101q^{58}+126q^{56}-95q^{54}+15q^{52}+89q^{50}-172q^{48}+207q^{46}-167q^{44}+71q^{42}+48q^{40}-152q^{38}+207q^{36}-190q^{34}+121q^{32}-15q^{30}-80q^{28}+140q^{26}-134q^{24}+74q^{22}+9q^{20}-79q^{18}+102q^{16}-75q^{14}+4q^{12}+79q^{10}-129q^{8}+135q^{6}-89q^{4}-q^{2}+85-143q^{-2}+151q^{-4}-113q^{-6}+48q^{-8}+25q^{-10}-78q^{-12}+103q^{-14}-94q^{-16}+64q^{-18}-24q^{-20}-12q^{-22}+33q^{-24}-41q^{-26}+36q^{-28}-23q^{-30}+12q^{-32}+q^{-34}-7q^{-36}+7q^{-38}-7q^{-40}+4q^{-42}-2q^{-44}+q^{-46}$

Further Quantum Invariants

K11a184 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["K11a184"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-11t^{2}+17t-19+17t^{-1}-11t^{-2}+5t^{-3}-t^{-4}$

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]=

{ 87, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-3q^{2}+5q-8+12q^{-1}-13q^{-2}+14q^{-3}-12q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+9a^{4}z^{4}-13a^{2}z^{4}+4z^{4}-3a^{6}z^{2}+12a^{4}z^{2}-11a^{2}z^{2}+4z^{2}-2a^{6}+4a^{4}-2a^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+6a^{3}z^{9}+3az^{9}+4a^{6}z^{8}+5a^{4}z^{8}+5a^{2}z^{8}+4z^{8}+4a^{7}z^{7}-4a^{5}z^{7}-17a^{3}z^{7}-6az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-6a^{6}z^{6}-21a^{4}z^{6}-26a^{2}z^{6}+z^{6}a^{-2}-13z^{6}+a^{9}z^{5}-6a^{7}z^{5}+3a^{5}z^{5}+18a^{3}z^{5}-2az^{5}-10z^{5}a^{-1}-6a^{8}z^{4}+4a^{6}z^{4}+34a^{4}z^{4}+39a^{2}z^{4}-3z^{4}a^{-2}+12z^{4}-2a^{9}z^{3}-2a^{3}z^{3}+7az^{3}+7z^{3}a^{-1}+2a^{8}z^{2}-5a^{6}z^{2}-20a^{4}z^{2}-20a^{2}z^{2}+z^{2}a^{-2}-6z^{2}+a^{9}z-a^{5}z-a^{3}z-2az-za^{-1}+2a^{6}+4a^{4}+2a^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_112,}

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["K11a184"];

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]=

{ $-t^{4}+5t^{3}-11t^{2}+17t-19+17t^{-1}-11t^{-2}+5t^{-3}-t^{-4}$, $q^{3}-3q^{2}+5q-8+12q^{-1}-13q^{-2}+14q^{-3}-12q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$ }

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_112,}

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, -4)

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 $8$ $-32$ $32$ ${\frac {364}{3}}$ ${\frac {68}{3}}$ $-256$ $-{\frac {1472}{3}}$ $-{\frac {128}{3}}$ $-96$ ${\frac {256}{3}}$ $512$ ${\frac {2912}{3}}$ ${\frac {544}{3}}$ ${\frac {33511}{15}}$ ${\frac {596}{15}}$ ${\frac {36844}{45}}$ ${\frac {473}{9}}$ ${\frac {1351}{15}}$

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 $s=$-2 is the signature of K11a184. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a183 K11a185