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

K11n151 K11n153

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8493 X12,5,13,6 X2837 X9,19,10,18 X11,20,12,21 X4,13,5,14 X15,11,16,10 X17,1,18,22 X19,14,20,15 X21,17,22,16

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 8 12 2 -18 -20 4 -10 -22 -14 -16

Alexander Polynomial:

- 2t-2 + 6t-1 - 7 + 6t - 2t2

Conway Polynomial:

1 - 2z2 - 2z4

Other knots with the same Alexander/Conway Polynomial:

{86, K11n20, K11n151, ...}

Determinant and Signature:

{23, 2}

Jones Polynomial:

q-2 - q-1 + 1 + q - 3q2 + 4q3 - 5q4 + 5q5 - 4q6 + 3q7 - q8

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

{K11n151, ...}

A2 (sl(3)) Invariant:

q-6 + q-4 + 2q-2 + q2 - 2q4 - 2q6 - 2q10 + 2q12 + 2q16 + q18 - q20 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + 5a-4 + 9a-4z2 + 5a-4z4 + a-4z6 - 7a-2 - 13a-2z2 - 7a-2z4 - a-2z6 + 4 + 4z2 + z4

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 3a-8z2 - 8a-8z4 + 3a-8z6 - 2a-7z + 4a-7z3 - 8a-7z5 + 3a-7z7 + a-6 + 2a-6z2 - 5a-6z4 + a-6z8 - 6a-5z + 15a-5z3 - 11a-5z5 + 3a-5z7 + 5a-4 - 19a-4z2 + 28a-4z4 - 13a-4z6 + 2a-4z8 - 4a-3z + 2a-3z3 + 11a-3z5 - 7a-3z7 + a-3z9 + 7a-2 - 31a-2z2 + 40a-2z4 - 17a-2z6 + 2a-2z8 - 7a-1z3 + 13a-1z5 - 7a-1z7 + a-1z9 + 4 - 13z2 + 15z4 - 7z6 + z8

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=2 is the signature of 11152. 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                     2   j = 13                   2 1   j = 11                 3 2     j = 9             1 3 2       j = 7             2 3         j = 5         1 3 3           j = 3       1 1 2             j = 1       1 3               j = -1   1 1                   j = -3                         j = -5 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, NonAlternating, 152]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 152]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[9, 19, 10, 18], X[11, 20, 12, 21], X[4, 13, 5, 14], X[15, 11, 16, 10], > X[17, 1, 18, 22], X[19, 14, 20, 15], X[21, 17, 22, 16]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 152]]
Out[4]=	GaussCode[1, -4, 2, -7, 3, -1, 4, -2, -5, 8, -6, -3, 7, 10, -8, 11, -9, 5, -10, > 6, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 152]]
Out[5]=	DTCode[6, 8, 12, 2, -18, -20, 4, -10, -22, -14, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 152]][t]
Out[6]=	2 6 2 -7 - -- + - + 6 t - 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 152]][z]
Out[7]=	2 4 1 - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 6], Knot[11, NonAlternating, 20], Knot[11, NonAlternating, 151], > Knot[11, NonAlternating, 152]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 152]], KnotSignature[Knot[11, NonAlternating, 152]]}
Out[9]=	{23, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 152]][q]
Out[10]=	-2 1 2 3 4 5 6 7 8 1 + q - - + q - 3 q + 4 q - 5 q + 5 q - 4 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 151], Knot[11, NonAlternating, 152]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 152]][q]
Out[12]=	-6 -4 2 2 4 6 10 12 16 18 20 22 q + q + -- + q - 2 q - 2 q - 2 q + 2 q + 2 q + q - q + q - 2 q 24 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 152]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 -6 5 7 2 2 z 9 z 13 z 4 z 5 z 7 z z z 4 - a + -- - -- + 4 z - ---- + ---- - ----- + z - -- + ---- - ---- + -- - -- 4 2 6 4 2 6 4 2 4 2 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 152]][a, z]
Out[14]=	2 2 2 2 -6 5 7 2 z 6 z 4 z 2 3 z 2 z 19 z 31 z 4 + a + -- + -- - --- - --- - --- - 13 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 4 4 2 z 4 z 15 z 2 z 7 z 4 8 z 5 z 28 z 40 z > ---- + ---- + ----- + ---- - ---- + 15 z - ---- - ---- + ----- + ----- + 9 7 5 3 a 8 6 4 2 a a a a a a a a 5 5 5 5 5 6 6 6 7 z 8 z 11 z 11 z 13 z 6 3 z 13 z 17 z 3 z > -- - ---- - ----- + ----- + ----- - 7 z + ---- - ----- - ----- + ---- + 9 7 5 3 a 8 4 2 7 a a a a a a a a 7 7 7 8 8 8 9 9 3 z 7 z 7 z 8 z 2 z 2 z z z > ---- - ---- - ---- + z + -- + ---- + ---- + -- + -- 5 3 a 6 4 2 3 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 152]], Vassiliev[3][Knot[11, NonAlternating, 152]]}
Out[15]=	{-2, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 152]][q, t]
Out[16]=	3 3 5 1 1 1 q q 3 5 5 2 3 q + q + q + ----- + ---- + ---- + - + -- + 2 q t + 3 q t + 3 q t + 5 4 3 2 t t q t q t q t 7 2 9 2 7 3 9 3 9 4 11 4 11 5 > 2 q t + q t + 3 q t + 3 q t + 2 q t + 3 q t + 2 q t + 13 5 13 6 15 6 17 7 > 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n152

K11n151 K11n153