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DrawMorseLink

PD Presentation:

X12,1,13,2 X8493 X18,14,19,13 X22,20,11,19 X20,7,21,8 X6,21,7,22 X4,15,5,16 X14,5,15,6 X16,10,17,9 X2,11,3,12 X10,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 3q-9/2 - 8q-7/2 + 12q-5/2 - 18q-3/2 + 21q-1/2 - 21q1/2 + 19q3/2 - 15q5/2 + 9q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + q-16 - q-14 + 3q-12 + 2q-10 + 6q-6 - 2q-4 - 1 - 4q2 + 4q4 - 3q6 + 4q8 + q10 - 3q12 + 2q14 - q16

HOMFLY-PT Polynomial:

2a-3z + 2a-3z3 + a-3z5 - 6a-1z - 8a-1z3 - 4a-1z5 - a-1z7 + 6az + 8az3 + 3az5 - a3z-1 - 5a3z - 3a3z3 + a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 + a-5z3 - 4a-5z5 - 2a-4z2 + 7a-4z4 - 9a-4z6 + 2a-3z - 11a-3z3 + 20a-3z5 - 14a-3z7 + 2a-2z2 - 10a-2z4 + 22a-2z6 - 14a-2z8 + 5a-1z - 21a-1z3 + 26a-1z5 + a-1z7 - 8a-1z9 + 11z2 - 40z4 + 53z6 - 17z8 - 2z10 + 4az - 14az3 - az5 + 25az7 - 12az9 + 9a2z2 - 32a2z4 + 32a2z6 - 6a2z8 - 2a2z10 - a3z-1 + 5a3z - 11a3z3 + a3z5 + 9a3z7 - 4a3z9 + a4 + 2a4z2 - 10a4z4 + 10a4z6 - 3a4z8 - a5z-1 + 4a5z - 6a5z3 + 4a5z5 - a5z7

Khovanov Homology:

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 = 12                       1 j = 10                     3   j = 8                   6 1   j = 6                 9 3     j = 4               10 6       j = 2             11 9         j = 0           11 11           j = -2         7 10             j = -4       5 11               j = -6     3 7                 j = -8   1 6                   j = -10   2                     j = -12 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]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 350]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 350]]
Out[4]=	PD[X[12, 1, 13, 2], X[8, 4, 9, 3], X[18, 14, 19, 13], X[22, 20, 11, 19], > X[20, 7, 21, 8], X[6, 21, 7, 22], X[4, 15, 5, 16], X[14, 5, 15, 6], > X[16, 10, 17, 9], X[2, 11, 3, 12], X[10, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -7, 8, -6, 5, -2, 9, -11}, > {10, -1, 3, -8, 7, -9, 11, -3, 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 8 12 18 21 3/2 -q + ---- - ---- + ---- - ---- + ------- - 21 Sqrt[q] + 19 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 15 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -14 3 2 6 2 2 4 6 8 -1 + q + q - q + --- + --- + -- - -- - 4 q + 4 q - 3 q + 4 q + 12 10 6 4 q q q q 10 12 14 16 > q - 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 350]][a, z]
Out[8]=	3 5 3 3 a a 2 z 6 z 3 5 2 z 8 z 3 -(--) + -- + --- - --- + 6 a z - 5 a z + a z + ---- - ---- + 8 a z - z z 3 a 3 a a a 5 5 7 3 3 z 4 z 5 z > 3 a z + -- - ---- + 3 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 350]][a, z]
Out[9]=	3 5 2 2 4 a a 2 z 5 z 3 5 2 2 z 2 z a - -- - -- + --- + --- + 4 a z + 5 a z + 4 a z + 11 z - ---- + ---- + z z 3 a 4 2 a a a 3 3 3 2 2 4 2 z 11 z 21 z 3 3 3 5 3 > 9 a z + 2 a z + -- - ----- - ----- - 14 a z - 11 a z - 6 a z - 5 3 a a a 4 4 4 5 5 5 4 z 7 z 10 z 2 4 4 4 4 z 20 z 26 z > 40 z - -- + ---- - ----- - 32 a z - 10 a z - ---- + ----- + ----- - 6 4 2 5 3 a a a a a a 6 6 5 3 5 5 5 6 9 z 22 z 2 6 4 6 > a z + a z + 4 a z + 53 z - ---- + ----- + 32 a z + 10 a z - 4 2 a a 7 7 8 14 z z 7 3 7 5 7 8 14 z 2 8 > ----- + -- + 25 a z + 9 a z - a z - 17 z - ----- - 6 a z - 3 a 2 a a 9 4 8 8 z 9 3 9 10 2 10 > 3 a z - ---- - 12 a z - 4 a z - 2 z - 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 6 3 7 5 11 11 + 11 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 7 11 10 2 4 4 2 6 2 6 3 > ----- + -- + ---- + 9 q t + 10 q t + 6 q t + 9 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a350

L11a349 L11a351