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DrawMorseLink

PD Presentation:

X6172 X3,15,4,14 X9,22,10,5 X7,19,8,18 X17,9,18,8 X19,13,20,12 X11,21,12,20 X15,10,16,11 X21,16,22,17 X2536 X13,1,14,4

Gauss Code:

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

Jones Polynomial:

q-7/2 - 4q-5/2 + 6q-3/2 - 9q-1/2 + 9q1/2 - 9q3/2 + 8q5/2 - 6q7/2 + 3q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-12 + 3q-8 + 3q-4 + 2q-2 + 2q2 - 2q4 + q6 - q8 - q10 + 2q12 - q14 + q16 + q18

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z + 3a-3z-1 + 6a-3z + 3a-3z3 - 4a-1z-1 - 9a-1z - 7a-1z3 - 2a-1z5 + 2az-1 + 5az + 3az3 - a3z

Kauffman Polynomial:

- a-5z-1 + 4a-5z - 6a-5z3 + 4a-5z5 - a-5z7 - a-4 + 6a-4z2 - 14a-4z4 + 12a-4z6 - 3a-4z8 - 3a-3z-1 + 18a-3z - 34a-3z3 + 21a-3z5 + a-3z7 - 2a-3z9 - 3a-2 + 17a-2z2 - 42a-2z4 + 39a-2z6 - 10a-2z8 - 4a-1z-1 + 26a-1z - 56a-1z3 + 42a-1z5 - 5a-1z7 - 2a-1z9 - 2 + 12z2 - 27z4 + 25z6 - 7z8 - 2az-1 + 15az - 32az3 + 25az5 - 7az7 - a2 + a2z4 - 2a2z6 + 3a3z - 4a3z3 - a4z2

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                   1 j = 10                 2   j = 8               4 1   j = 6             4 2     j = 4           5 4       j = 2       1 5 4         j = 0       5 5           j = -2     3 6             j = -4   1 3               j = -6   3                 j = -8 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, NonAlternating, 124]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 124]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 15, 4, 14], X[9, 22, 10, 5], X[7, 19, 8, 18], > X[17, 9, 18, 8], X[19, 13, 20, 12], X[11, 21, 12, 20], X[15, 10, 16, 11], > X[21, 16, 22, 17], X[2, 5, 3, 6], X[13, 1, 14, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -4, 5, -3, 8, -7, 6, -11, 2, -8, 9, -5, 4, > -6, 7, -9, 3}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 6 9 3/2 5/2 7/2 q - ---- + ---- - ------- + 9 Sqrt[q] - 9 q + 8 q - 6 q + 5/2 3/2 Sqrt[q] q q 9/2 11/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 3 3 2 2 4 6 8 10 12 14 16 18 -q + -- + -- + -- + 2 q - 2 q + q - q - q + 2 q - q + q + q 8 4 2 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 124]][a, z]
Out[8]=	3 3 1 3 4 2 a z 6 z 9 z 3 3 z 7 z -(----) + ---- - --- + --- - -- + --- - --- + 5 a z - a z + ---- - ---- + 5 3 a z z 5 3 a 3 a a z a z a a a 5 3 2 z > 3 a z - ---- a
In[9]:=	Kauffman[Link[11, NonAlternating, 124]][a, z]
Out[9]=	-4 3 2 1 3 4 2 a 4 z 18 z 26 z -2 - a - -- - a - ---- - ---- - --- - --- + --- + ---- + ---- + 15 a z + 2 5 3 a z z 5 3 a a a z a z a a 2 2 3 3 3 3 2 6 z 17 z 4 2 6 z 34 z 56 z 3 > 3 a z + 12 z + ---- + ----- - a z - ---- - ----- - ----- - 32 a z - 4 2 5 3 a a a a a 4 4 5 5 5 3 3 4 14 z 42 z 2 4 4 z 21 z 42 z 5 > 4 a z - 27 z - ----- - ----- + a z + ---- + ----- + ----- + 25 a z + 4 2 5 3 a a a a a 6 6 7 7 7 8 6 12 z 39 z 2 6 z z 5 z 7 8 3 z > 25 z + ----- + ----- - 2 a z - -- + -- - ---- - 7 a z - 7 z - ---- - 4 2 5 3 a 4 a a a a a 8 9 9 10 z 2 z 2 z > ----- - ---- - ---- 2 3 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	6 2 1 3 1 3 3 2 2 2 5 + -- + q + ----- + ----- + ----- + ---- + ---- + 5 t + 5 q t + 4 q t + 2 8 3 6 2 4 2 4 2 q q t q t q t q t q t 4 2 4 3 6 3 6 4 8 4 8 5 10 5 12 6 > 5 q t + 4 q t + 4 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n124

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