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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X9,16,10,17 X2,14,3,13

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-7 + q-6 - 2q-5 + 3q-4 - 2q-3 + 3q-2 - q-1 + 3

A2 (sl(3)) Invariant:

- q-22 - q-20 - 2q-18 - 3q-16 - q-14 + 3q-10 + 5q-8 + 6q-6 + 7q-4 + 6q-2 + 5 + 3q2

HOMFLY-PT Polynomial:

2z-2 + 3 - 5a2z-2 - 8a2 - 3a2z2 + 4a4z-2 + 7a4 + 4a4z2 + a4z4 - a6z-2 - 2a6 - a6z2

Kauffman Polynomial:

- 2z-2 + 5 + 5az-1 - 8az + 3az3 - 5a2z-2 + 11a2 - 7a2z2 - a2z4 + a2z6 + 9a3z-1 - 21a3z + 19a3z3 - 10a3z5 + 2a3z7 - 4a4z-2 + 10a4 - 11a4z2 + 6a4z4 - 4a4z6 + a4z8 + 5a5z-1 - 19a5z + 27a5z3 - 16a5z5 + 3a5z7 - a6z-2 + 3a6 - 4a6z2 + 7a6z4 - 5a6z6 + a6z8 + a7z-1 - 6a7z + 11a7z3 - 6a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = 1               3 j = -1             2 4 j = -3           1   1 j = -5         1 2     j = -7       2 1       j = -9       1         j = -11   1 2           j = -13                 j = -15 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, 378]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 378]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[15, 1, 16, 4], X[5, 10, 6, 11], > X[3, 8, 4, 9], X[22, 18, 19, 17], X[11, 20, 12, 21], X[19, 12, 20, 13], > X[18, 22, 5, 21], X[9, 16, 10, 17], X[2, 14, 3, 13]]
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]=	-7 -6 2 3 2 3 1 3 - q + q - -- + -- - -- + -- - - 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 2 3 -14 3 5 6 7 6 2 5 - q - q - --- - --- - q + --- + -- + -- + -- + -- + 3 q 18 16 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 378]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 4 2 6 2 3 - 8 a + 7 a - 2 a + -- - ---- + ---- - -- - 3 a z + 4 a z - a z + 2 2 2 2 z z z z 4 4 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 378]][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 5 + 11 a + 10 a + 3 a - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 2 2 2 2 z z z z z z z z 3 5 7 2 2 4 2 6 2 > 8 a z - 21 a z - 19 a z - 6 a z - 7 a z - 11 a z - 4 a z + 3 3 3 5 3 7 3 2 4 4 4 6 4 > 3 a z + 19 a z + 27 a z + 11 a z - a z + 6 a z + 7 a z - 3 5 5 5 7 5 2 6 4 6 6 6 3 7 > 10 a z - 16 a z - 6 a z + a z - 4 a z - 5 a z + 2 a z + 5 7 7 7 4 8 6 8 > 3 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-3 4 1 1 2 1 2 1 1 q + - + 3 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 11 6 11 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 2 1 2 > ----- + ----- + --- 5 2 3 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n378

L11n377 L11n379