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

K11a171 K11a173

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 22 14 2 20 8 6 12 16

Alexander Polynomial:

2t-3 - 13t-2 + 33t-1 - 43 + 33t - 13t2 + 2t3

Conway Polynomial:

1 - z2 - z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a54, ...}

Determinant and Signature:

{139, 2}

Jones Polynomial:

q-3 - 3q-2 + 8q-1 - 14 + 19q - 22q2 + 23q3 - 20q4 + 15q5 - 9q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 4q-4 - 2q-2 - 1 + 3q2 - 5q4 + 3q6 - 2q8 + q10 + 3q12 - 3q14 + 5q16 - 2q18 - 2q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6 - a-6z2 - a-6z4 + 3a-4 + 4a-4z2 + 2a-4z4 + a-4z6 - 2a-2 - 3a-2z2 + a-2z6 - 2z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 5a-8z4 + 4a-8z6 - a-7z + 6a-7z3 - 12a-7z5 + 8a-7z7 + a-6 - 4a-6z2 + 11a-6z4 - 16a-6z6 + 10a-6z8 - a-5z + 8a-5z3 - 9a-5z5 - 3a-5z7 + 7a-5z9 + 3a-4 - 15a-4z2 + 35a-4z4 - 39a-4z6 + 15a-4z8 + 2a-4z10 + a-3z - 3a-3z3 + 8a-3z5 - 19a-3z7 + 12a-3z9 + 2a-2 - 12a-2z2 + 25a-2z4 - 29a-2z6 + 10a-2z8 + 2a-2z10 + a-1z3 - 3a-1z5 - 5a-1z7 + 5a-1z9 + z2 + 3z4 - 9z6 + 5z8 - az + 5az3 - 7az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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

{-1, 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=2 is the signature of 11172. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   6 1   j = 11                 9 3     j = 9               11 6       j = 7             12 9         j = 5           10 11           j = 3         9 12             j = 1       6 11               j = -1     2 8                 j = -3   1 6                   j = -5   2                     j = -7 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, 172]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 172]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[22, 8, 1, 7], > X[14, 10, 15, 9], X[2, 11, 3, 12], X[20, 14, 21, 13], X[8, 16, 9, 15], > X[6, 17, 7, 18], X[12, 20, 13, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 172]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -9, 4, -8, 5, -2, 6, -10, 7, -5, 8, -11, 9, -3, 10, > -7, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 172]]
Out[5]=	DTCode[4, 10, 18, 22, 14, 2, 20, 8, 6, 12, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 172]][t]
Out[6]=	2 13 33 2 3 -43 + -- - -- + -- + 33 t - 13 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 172]][z]
Out[7]=	2 4 6 1 - z - z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 54], Knot[11, Alternating, 172]}
In[9]:=	{KnotDet[Knot[11, Alternating, 172]], KnotSignature[Knot[11, Alternating, 172]]}
Out[9]=	{139, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 172]][q]
Out[10]=	-3 3 8 2 3 4 5 6 7 8 -14 + q - -- + - + 19 q - 22 q + 23 q - 20 q + 15 q - 9 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 172]}
In[12]:=	A2Invariant[Knot[11, Alternating, 172]][q]
Out[12]=	-10 -6 4 2 2 4 6 8 10 12 14 -1 + q - q + -- - -- + 3 q - 5 q + 3 q - 2 q + q + 3 q - 3 q + 4 2 q q 16 18 20 22 24 > 5 q - 2 q - 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 172]][a, z]
Out[13]=	2 2 2 4 4 6 -6 3 2 2 2 z 4 z 3 z 2 2 4 z 2 z z -a + -- - -- + a - 2 z - -- + ---- - ---- + a z - 2 z - -- + ---- + -- + 4 2 6 4 2 6 4 4 a a a a a a a a 6 z > -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 172]][a, z]
Out[14]=	2 2 2 2 -6 3 2 2 z z z 2 z 4 z 15 z 12 z a + -- + -- - a - -- - -- + -- - a z + z + -- - ---- - ----- - ----- + 4 2 7 5 3 8 6 4 2 a a a a a a a a a 3 3 3 3 3 4 4 2 2 z 6 z 8 z 3 z z 3 4 5 z 11 z > 3 a z - -- + ---- + ---- - ---- + -- + 5 a z + 3 z - ---- + ----- + 9 7 5 3 a 8 6 a a a a a a 4 4 5 5 5 5 5 35 z 25 z 2 4 z 12 z 9 z 8 z 3 z 5 6 > ----- + ----- - 3 a z + -- - ----- - ---- + ---- - ---- - 7 a z - 9 z + 4 2 9 7 5 3 a a a a a a a 6 6 6 6 7 7 7 7 4 z 16 z 39 z 29 z 2 6 8 z 3 z 19 z 5 z > ---- - ----- - ----- - ----- + a z + ---- - ---- - ----- - ---- + 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 7 8 10 z 15 z 10 z 7 z 12 z 5 z 2 z 2 z > 3 a z + 5 z + ----- + ----- + ----- + ---- + ----- + ---- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 172]], Vassiliev[3][Knot[11, Alternating, 172]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, Alternating, 172]][q, t]
Out[16]=	3 1 2 1 6 2 8 6 q 3 11 q + 9 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 12 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 10 q t + 11 q t + 12 q t + 9 q t + 11 q t + 6 q t + 9 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a172

K11a171 K11a173