© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n181 K11n183

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,12,4,13 X16,5,17,6 X18,8,19,7 X14,10,15,9 X11,4,12,5 X20,13,21,14 X22,16,1,15 X2,17,3,18 X8,20,9,19 X10,21,11,22

Gauss Code:

{1, -9, -2, 6, 3, -1, 4, -10, 5, -11, -6, 2, 7, -5, 8, -3, 9, -4, 10, -7, 11, -8}

DT (Dowker-Thistlethwaite) Code:

6 -12 16 18 14 -4 20 22 2 8 10

Alexander Polynomial:

t-4 - 5t-3 + 11t-2 - 18t-1 + 23 - 18t + 11t2 - 5t3 + t4

Conway Polynomial:

1 - 3z2 + z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{93, 0}

Jones Polynomial:

3q-4 - 7q-3 + 11q-2 - 15q-1 + 16 - 15q + 13q2 - 8q3 + 4q4 - q5

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-16 + 3q-12 - 2q-10 + q-8 - q-6 - 4q-4 + 3q-2 - 4 + 4q2 - q4 + q6 + 3q8 - 2q10 + 2q12 - q14

HOMFLY-PT Polynomial:

a-2 - 2a-2z2 - 3a-2z4 - a-2z6 + 1 + 4z2 + 8z4 + 5z6 + z8 - 3a2 - 6a2z2 - 4a2z4 - a2z6 + 2a4 + a4z2

Kauffman Polynomial:

- a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 4a-4z6 - a-3z + 4a-3z3 - 11a-3z5 + 7a-3z7 - a-2 + 2a-2z2 - 9a-2z6 + 7a-2z8 - 2a-1z + 12a-1z3 - 19a-1z5 + 6a-1z7 + 3a-1z9 + 1 - 9z2 + 23z4 - 27z6 + 13z8 + 2az + 2az3 - 7az5 + 2az7 + 3az9 + 3a2 - 18a2z2 + 23a2z4 - 14a2z6 + 6a2z8 + 3a3z - 5a3z3 + 3a3z7 + 2a4 - 9a4z2 + 6a4z4

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-3, 2}

Khovanov Homology: (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 11182. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                   1 j = 9                 3   j = 7               5 1   j = 5             8 3     j = 3           7 5       j = 1         9 8         j = -1       7 8           j = -3     4 8             j = -5   3 7               j = -7   4                 j = -9 3

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 182]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 182]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 12, 4, 13], X[16, 5, 17, 6], X[18, 8, 19, 7], > X[14, 10, 15, 9], X[11, 4, 12, 5], X[20, 13, 21, 14], X[22, 16, 1, 15], > X[2, 17, 3, 18], X[8, 20, 9, 19], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 182]]
Out[4]=	GaussCode[1, -9, -2, 6, 3, -1, 4, -10, 5, -11, -6, 2, 7, -5, 8, -3, 9, -4, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 182]]
Out[5]=	DTCode[6, -12, 16, 18, 14, -4, 20, 22, 2, 8, 10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 182]][t]
Out[6]=	-4 5 11 18 2 3 4 23 + t - -- + -- - -- - 18 t + 11 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 182]][z]
Out[7]=	2 4 6 8 1 - 3 z + z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 182]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 182]], KnotSignature[Knot[11, NonAlternating, 182]]}
Out[9]=	{93, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 182]][q]
Out[10]=	3 7 11 15 2 3 4 5 16 + -- - -- + -- - -- - 15 q + 13 q - 8 q + 4 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 182]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 182]][q]
Out[12]=	-16 3 2 -8 -6 4 3 2 4 6 8 10 -4 + q + --- - --- + q - q - -- + -- + 4 q - q + q + 3 q - 2 q + 12 10 4 2 q q q q 12 14 > 2 q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 182]][a, z]
Out[13]=	2 4 -2 2 4 2 2 z 2 2 4 2 4 3 z 2 4 1 + a - 3 a + 2 a + 4 z - ---- - 6 a z + a z + 8 z - ---- - 4 a z + 2 2 a a 6 6 z 2 6 8 > 5 z - -- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 182]][a, z]
Out[14]=	2 2 -2 2 4 z 2 z 3 2 2 z 2 z 1 - a + 3 a + 2 a - -- - --- + 2 a z + 3 a z - 9 z + ---- + ---- - 3 a 4 2 a a a 3 3 3 4 2 2 4 2 z 4 z 12 z 3 3 3 4 6 z > 18 a z - 9 a z - -- + ---- + ----- + 2 a z - 5 a z + 23 z - ---- + 5 3 a 4 a a a 5 5 5 6 6 2 4 4 4 z 11 z 19 z 5 6 4 z 9 z > 23 a z + 6 a z + -- - ----- - ----- - 7 a z - 27 z + ---- - ---- - 5 3 a 4 2 a a a a 7 7 8 9 2 6 7 z 6 z 7 3 7 8 7 z 2 8 3 z > 14 a z + ---- + ---- + 2 a z + 3 a z + 13 z + ---- + 6 a z + ---- + 3 a 2 a a a 9 > 3 a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 182]], Vassiliev[3][Knot[11, NonAlternating, 182]]}
Out[15]=	{-3, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 182]][q, t]
Out[16]=	8 3 4 3 7 4 8 7 3 - + 9 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 8 q t + 7 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 11 5 > 5 q t + 8 q t + 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n182

K11n181 K11n183