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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X22,18,19,17 X20,12,21,11 X12,20,13,19 X18,22,5,21 X9,16,10,17 X13,2,14,3

Gauss Code:

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

Jones Polynomial:

- q-4 + q-3 + q-2 + q-1 + 1 + q2

A2 (sl(3)) Invariant:

- q-20 - q-18 - q-16 - 2q-14 - q-12 + q-10 + 4q-8 + 6q-6 + 6q-4 + 6q-2 + 4 + 3q2 + 2q4 + q6

HOMFLY-PT Polynomial:

2z-2 + 6 + 5z2 + z4 - 5a2z-2 - 13a2 - 15a2z2 - 7a2z4 - a2z6 + 4a4z-2 + 8a4 + 6a4z2 + a4z4 - a6z-2 - a6

Kauffman Polynomial:

- 2z-2 + 11 - 22z2 + 21z4 - 8z6 + z8 + 5az-1 - 18az + 20az3 - 8az5 + az7 - 5a2z-2 + 21a2 - 37a2z2 + 28a2z4 - 9a2z6 + a2z8 + 9a3z-1 - 27a3z + 26a3z3 - 9a3z5 + a3z7 - 4a4z-2 + 12a4 - 15a4z2 + 7a4z4 - a4z6 + 5a5z-1 - 9a5z + 6a5z3 - a5z5 - a6z-2 + a6 + a7z-1

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 5                 1 j = 3                 1 j = 1             1     j = -1         2         j = -3         3 1       j = -5     2             j = -7     1 1           j = -9 1 2               j = -11 1 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 379]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 379]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 10, 6, 11], > X[3, 8, 4, 9], X[22, 18, 19, 17], X[20, 12, 21, 11], X[12, 20, 13, 19], > X[18, 22, 5, 21], X[9, 16, 10, 17], X[13, 2, 14, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {8, -7, 9, -6}, > {-4, -1, 2, 5, -10, 4, 7, -8, -11, -2, 3, 10, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-4 -3 -2 1 2 1 - q + q + q + - + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 2 -12 -10 4 6 6 6 2 4 4 - q - q - q - --- - q + q + -- + -- + -- + -- + 3 q + 2 q + 14 8 6 4 2 q q q q q 6 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 379]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 2 4 2 6 - 13 a + 8 a - a + -- - ---- + ---- - -- + 5 z - 15 a z + 6 a z + 2 2 2 2 z z z z 4 2 4 4 4 2 6 > z - 7 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 379]][a, z]
Out[9]=	2 4 6 3 5 7 2 4 6 2 5 a 4 a a 5 a 9 a 5 a a 11 + 21 a + 12 a + a - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 2 2 2 2 z z z z z z z z 3 5 2 2 2 4 2 3 > 18 a z - 27 a z - 9 a z - 22 z - 37 a z - 15 a z + 20 a z + 3 3 5 3 4 2 4 4 4 5 3 5 > 26 a z + 6 a z + 21 z + 28 a z + 7 a z - 8 a z - 9 a z - 5 5 6 2 6 4 6 7 3 7 8 2 8 > a z - 8 z - 9 a z - a z + a z + a z + z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 2 1 1 1 2 1 2 1 t 2 -- + - + ------ + ----- + ------ + ----- + ----- + ----- + ---- + -- + q t + 3 q 11 4 9 4 11 3 9 3 7 2 5 2 7 3 q q t q t q t q t q t q t q t q 3 4 5 4 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n379

L11n378 L11n380