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

K11a169 K11a171

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 20 2 16 6 8 22 14 12

Alexander Polynomial:

t-4 - 6t-3 + 20t-2 - 40t-1 + 51 - 40t + 20t2 - 6t3 + t4

Conway Polynomial:

1 + 2z2 + 4z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{185, 0}

Jones Polynomial:

- q-5 + 4q-4 - 10q-3 + 18q-2 - 25q-1 + 30 - 30q + 27q2 - 20q3 + 13q4 - 6q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-14 + 2q-12 - 4q-10 + 3q-8 + 2q-6 - 5q-4 + 6q-2 - 5 + 4q2 + q4 - q6 + 6q8 - 5q10 + 2q12 - 3q16 + q18

HOMFLY-PT Polynomial:

- a-4 + a-4z4 + a-2 - 3a-2z2 - 5a-2z4 - 2a-2z6 + 2 + 9z2 + 11z4 + 5z6 + z8 - a2 - 4a2z2 - 3a2z4 - a2z6

Kauffman Polynomial:

a-6z6 + 2a-5z - 8a-5z5 + 6a-5z7 - a-4 + 2a-4z2 + 11a-4z4 - 26a-4z6 + 13a-4z8 + 2a-3z + 4a-3z3 - 7a-3z5 - 15a-3z7 + 12a-3z9 - a-2 - 6a-2z2 + 41a-2z4 - 65a-2z6 + 22a-2z8 + 4a-2z10 - 2a-1z + 10a-1z3 + 3a-1z5 - 37a-1z7 + 23a-1z9 + 2 - 16z2 + 51z4 - 62z6 + 22z8 + 4z10 - 4az + 15az3 - 11az5 - 7az7 + 11az9 + a2 - 7a2z2 + 17a2z4 - 20a2z6 + 13a2z8 - 2a3z + 8a3z3 - 12a3z5 + 9a3z7 + a4z2 - 4a4z4 + 4a4z6 - a5z3 + a5z5

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

{2, 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 11170. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     5   j = 9                   8 1   j = 7                 12 5     j = 5               15 8       j = 3             15 12         j = 1           15 15           j = -1         11 16             j = -3       7 14               j = -5     3 11                 j = -7   1 7                   j = -9   3                     j = -11 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, 170]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 170]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[18, 5, 19, 6], X[20, 8, 21, 7], > X[2, 10, 3, 9], X[16, 11, 17, 12], X[6, 14, 7, 13], X[8, 15, 9, 16], > X[22, 17, 1, 18], X[14, 19, 15, 20], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 170]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -8, 5, -2, 6, -11, 7, -10, 8, -6, 9, -3, 10, > -4, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 170]]
Out[5]=	DTCode[4, 10, 18, 20, 2, 16, 6, 8, 22, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 170]][t]
Out[6]=	-4 6 20 40 2 3 4 51 + t - -- + -- - -- - 40 t + 20 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 170]][z]
Out[7]=	2 4 6 8 1 + 2 z + 4 z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 170]}
In[9]:=	{KnotDet[Knot[11, Alternating, 170]], KnotSignature[Knot[11, Alternating, 170]]}
Out[9]=	{185, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 170]][q]
Out[10]=	-5 4 10 18 25 2 3 4 5 6 30 - q + -- - -- + -- - -- - 30 q + 27 q - 20 q + 13 q - 6 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 170]}
In[12]:=	A2Invariant[Knot[11, Alternating, 170]][q]
Out[12]=	-14 2 4 3 2 5 6 2 4 6 8 10 -5 - q + --- - --- + -- + -- - -- + -- + 4 q + q - q + 6 q - 5 q + 12 10 8 6 4 2 q q q q q q 12 16 18 > 2 q - 3 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 170]][a, z]
Out[13]=	2 4 4 -4 -2 2 2 3 z 2 2 4 z 5 z 2 4 2 - a + a - a + 9 z - ---- - 4 a z + 11 z + -- - ---- - 3 a z + 2 4 2 a a a 6 6 2 z 2 6 8 > 5 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 170]][a, z]
Out[14]=	2 2 -4 -2 2 2 z 2 z 2 z 3 2 2 z 6 z 2 - a - a + a + --- + --- - --- - 4 a z - 2 a z - 16 z + ---- - ---- - 5 3 a 4 2 a a a a 3 3 2 2 4 2 4 z 10 z 3 3 3 5 3 4 > 7 a z + a z + ---- + ----- + 15 a z + 8 a z - a z + 51 z + 3 a a 4 4 5 5 5 11 z 41 z 2 4 4 4 8 z 7 z 3 z 5 > ----- + ----- + 17 a z - 4 a z - ---- - ---- + ---- - 11 a z - 4 2 5 3 a a a a a 6 6 6 7 3 5 5 5 6 z 26 z 65 z 2 6 4 6 6 z > 12 a z + a z - 62 z + -- - ----- - ----- - 20 a z + 4 a z + ---- - 6 4 2 5 a a a a 7 7 8 8 15 z 37 z 7 3 7 8 13 z 22 z 2 8 > ----- - ----- - 7 a z + 9 a z + 22 z + ----- + ----- + 13 a z + 3 a 4 2 a a a 9 9 10 12 z 23 z 9 10 4 z > ----- + ----- + 11 a z + 4 z + ----- 3 a 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 170]], Vassiliev[3][Knot[11, Alternating, 170]]}
Out[15]=	{2, 1}
In[16]:=	Kh[Knot[11, Alternating, 170]][q, t]
Out[16]=	16 1 3 1 7 3 11 7 14 -- + 15 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 11 3 3 2 5 2 5 3 7 3 > --- + 15 q t + 15 q t + 12 q t + 15 q t + 8 q t + 12 q t + q t 7 4 9 4 9 5 11 5 13 6 > 5 q t + 8 q t + q t + 5 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a170

K11a169 K11a171