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

K11a75 K11a77

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X18,9,19,10 X2,11,3,12 X20,14,21,13 X8,15,9,16 X22,18,1,17 X6,19,7,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 14 18 2 20 8 22 6 16

Alexander Polynomial:

t-4 - 6t-3 + 17t-2 - 30t-1 + 37 - 30t + 17t2 - 6t3 + t4

Conway Polynomial:

1 + z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a160, K11a289, ...}

Determinant and Signature:

{145, 0}

Jones Polynomial:

q-6 - 4q-5 + 9q-4 - 15q-3 + 20q-2 - 23q-1 + 24 - 20q + 15q2 - 9q3 + 4q4 - q5

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

{K11a160, K11a289, ...}

A2 (sl(3)) Invariant:

q-18 - q-16 + 2q-12 - 4q-10 + 3q-8 - 2q-6 - q-4 + 4q-2 - 3 + 6q2 - 3q4 + q6 + 2q8 - 3q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- a-2 - 3a-2z2 - 3a-2z4 - a-2z6 + 4 + 9z2 + 10z4 + 5z6 + z8 - 3a2 - 8a2z2 - 7a2z4 - 2a2z6 + a4 + 2a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + a-4z2 - 5a-4z4 + 4a-4z6 - a-3z + 6a-3z3 - 12a-3z5 + 8a-3z7 + a-2 - 5a-2z2 + 12a-2z4 - 16a-2z6 + 10a-2z8 - 2a-1z + 10a-1z3 - 11a-1z5 - 2a-1z7 + 7a-1z9 + 4 - 17z2 + 41z4 - 45z6 + 17z8 + 2z10 - 2az + 6az3 + az5 - 19az7 + 13az9 + 3a2 - 15a2z2 + 35a2z4 - 41a2z6 + 14a2z8 + 2a2z10 - 2a3z + 9a3z3 - 10a3z5 - 5a3z7 + 6a3z9 + a4 - 3a4z2 + 9a4z4 - 15a4z6 + 7a4z8 - a5z + 6a5z3 - 9a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

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

{0, 1}

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 1176. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 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                   6 1   j = 5                 9 3     j = 3               11 6       j = 1             13 9         j = -1           11 12           j = -3         9 12             j = -5       6 11               j = -7     3 9                 j = -9   1 6                   j = -11   3                     j = -13 1

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, Alternating, 76]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 76]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[20, 14, 21, 13], X[8, 15, 9, 16], > X[22, 18, 1, 17], X[6, 19, 7, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 76]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -4, 8, -11, 9, -5, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 76]]
Out[5]=	DTCode[4, 10, 12, 14, 18, 2, 20, 8, 22, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 76]][t]
Out[6]=	-4 6 17 30 2 3 4 37 + t - -- + -- - -- - 30 t + 17 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 76]][z]
Out[7]=	4 6 8 1 + z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 76], Knot[11, Alternating, 160], > Knot[11, Alternating, 289]}
In[9]:=	{KnotDet[Knot[11, Alternating, 76]], KnotSignature[Knot[11, Alternating, 76]]}
Out[9]=	{145, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 76]][q]
Out[10]=	-6 4 9 15 20 23 2 3 4 5 24 + q - -- + -- - -- + -- - -- - 20 q + 15 q - 9 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 76], Knot[11, Alternating, 160], > Knot[11, Alternating, 289]}
In[12]:=	A2Invariant[Knot[11, Alternating, 76]][q]
Out[12]=	-18 -16 2 4 3 2 -4 4 2 4 6 8 -3 + q - q + --- - --- + -- - -- - q + -- + 6 q - 3 q + q + 2 q - 12 10 8 6 2 q q q q q 10 12 14 > 3 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 76]][a, z]
Out[13]=	2 4 -2 2 4 2 3 z 2 2 4 2 4 3 z 4 - a - 3 a + a + 9 z - ---- - 8 a z + 2 a z + 10 z - ---- - 2 2 a a 6 2 4 4 4 6 z 2 6 8 > 7 a z + a z + 5 z - -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 76]][a, z]
Out[14]=	2 2 -2 2 4 z 2 z 3 5 2 z 5 z 4 + a + 3 a + a - -- - --- - 2 a z - 2 a z - a z - 17 z + -- - ---- - 3 a 4 2 a a a 3 3 3 2 2 4 2 6 2 z 6 z 10 z 3 3 3 > 15 a z - 3 a z + a z - -- + ---- + ----- + 6 a z + 9 a z + 5 3 a a a 4 4 5 5 3 4 5 z 12 z 2 4 4 4 6 4 z > 6 a z + 41 z - ---- + ----- + 35 a z + 9 a z - 2 a z + -- - 4 2 5 a a a 5 5 6 6 12 z 11 z 5 3 5 5 5 6 4 z 16 z > ----- - ----- + a z - 10 a z - 9 a z - 45 z + ---- - ----- - 3 a 4 2 a a a 7 7 2 6 4 6 6 6 8 z 2 z 7 3 7 5 7 > 41 a z - 15 a z + a z + ---- - ---- - 19 a z - 5 a z + 4 a z + 3 a a 8 9 8 10 z 2 8 4 8 7 z 9 3 9 10 > 17 z + ----- + 14 a z + 7 a z + ---- + 13 a z + 6 a z + 2 z + 2 a a 2 10 > 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 76]], Vassiliev[3][Knot[11, Alternating, 76]]}
Out[15]=	{0, 1}
In[16]:=	Kh[Knot[11, Alternating, 76]][q, t]
Out[16]=	12 1 3 1 6 3 9 6 11 -- + 13 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 9 12 11 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 11 q t + 6 q t + 9 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a76

K11a75 K11a77