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

K11n161 K11n163

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X12,6,13,5 X22,8,1,7 X16,10,17,9 X11,19,12,18 X8,14,9,13 X20,16,21,15 X17,4,18,5 X2,19,3,20 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 3t-2 + 14t-1 - 21 + 14t - 3t2

Conway Polynomial:

1 + 2z2 - 3z4

Other knots with the same Alexander/Conway Polynomial:

{939, ...}

Determinant and Signature:

{55, 2}

Jones Polynomial:

2q - 4q2 + 7q3 - 8q4 + 9q5 - 9q6 + 7q7 - 5q8 + 3q9 - q10

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

{...}

A2 (sl(3)) Invariant:

2q2 - q4 + 3q8 + 2q12 - 2q16 - q18 - 3q20 + 2q22 + q24 + 2q28 - q30 - q32

HOMFLY-PT Polynomial:

- a-10 + 3a-8 + 3a-8z2 - 4a-6 - 4a-6z2 - 2a-6z4 + 2a-4 + a-4z2 - a-4z4 + a-2 + 2a-2z2

Kauffman Polynomial:

- 2a-11z + 5a-11z3 - 4a-11z5 + a-11z7 + a-10 - 5a-10z2 + 16a-10z4 - 13a-10z6 + 3a-10z8 - 4a-9z + 14a-9z3 - 7a-9z5 - 4a-9z7 + 2a-9z9 + 3a-8 - 14a-8z2 + 32a-8z4 - 30a-8z6 + 8a-8z8 - 4a-7z + 12a-7z3 - 14a-7z5 + 2a-7z9 + 4a-6 - 14a-6z2 + 15a-6z4 - 14a-6z6 + 5a-6z8 - 2a-5z + 6a-5z3 - 10a-5z5 + 5a-5z7 + 2a-4 - 2a-4z2 - a-4z4 + 3a-4z6 + 3a-3z3 + a-3z5 - a-2 + 3a-2z2

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

{2, 4}

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 11162. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 21                   1 j = 19                 2   j = 17               3 1   j = 15             4 2     j = 13           5 3       j = 11         4 4         j = 9       4 5           j = 7     3 4             j = 5   1 4               j = 3 1 3                 j = 1 2

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, 162]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 162]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[12, 6, 13, 5], X[22, 8, 1, 7], > X[16, 10, 17, 9], X[11, 19, 12, 18], X[8, 14, 9, 13], X[20, 16, 21, 15], > X[17, 4, 18, 5], X[2, 19, 3, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 162]]
Out[4]=	GaussCode[1, -10, -2, 9, 3, -1, 4, -7, 5, 2, -6, -3, 7, -11, 8, -5, -9, 6, 10, > -8, 11, -4]
In[5]:=	DTCode[Knot[11, NonAlternating, 162]]
Out[5]=	DTCode[6, -10, 12, 22, 16, -18, 8, 20, -4, 2, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 162]][t]
Out[6]=	3 14 2 -21 - -- + -- + 14 t - 3 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 162]][z]
Out[7]=	2 4 1 + 2 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 39], Knot[11, NonAlternating, 162]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 162]], KnotSignature[Knot[11, NonAlternating, 162]]}
Out[9]=	{55, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 162]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 2 q - 4 q + 7 q - 8 q + 9 q - 9 q + 7 q - 5 q + 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 162]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 162]][q]
Out[12]=	2 4 8 12 16 18 20 22 24 28 30 32 2 q - q + 3 q + 2 q - 2 q - q - 3 q + 2 q + q + 2 q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 162]][a, z]
Out[13]=	2 2 2 2 4 4 -10 3 4 2 -2 3 z 4 z z 2 z 2 z z -a + -- - -- + -- + a + ---- - ---- + -- + ---- - ---- - -- 8 6 4 8 6 4 2 6 4 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 162]][a, z]
Out[14]=	2 2 2 -10 3 4 2 -2 2 z 4 z 4 z 2 z 5 z 14 z 14 z a + -- + -- + -- - a - --- - --- - --- - --- - ---- - ----- - ----- - 8 6 4 11 9 7 5 10 8 6 a a a a a a a a a a 2 2 3 3 3 3 3 4 4 4 2 z 3 z 5 z 14 z 12 z 6 z 3 z 16 z 32 z 15 z > ---- + ---- + ---- + ----- + ----- + ---- + ---- + ----- + ----- + ----- - 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 z 4 z 7 z 14 z 10 z z 13 z 30 z 14 z 3 z > -- - ---- - ---- - ----- - ----- + -- - ----- - ----- - ----- + ---- + 4 11 9 7 5 3 10 8 6 4 a a a a a a a a a a 7 7 7 8 8 8 9 9 z 4 z 5 z 3 z 8 z 5 z 2 z 2 z > --- - ---- + ---- + ---- + ---- + ---- + ---- + ---- 11 9 5 10 8 6 9 7 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 162]], Vassiliev[3][Knot[11, NonAlternating, 162]]}
Out[15]=	{2, 4}
In[16]:=	Kh[Knot[11, NonAlternating, 162]][q, t]
Out[16]=	3 3 5 5 2 7 2 7 3 9 3 9 4 2 q + q + 3 q t + q t + 4 q t + 3 q t + 4 q t + 4 q t + 5 q t + 11 4 11 5 13 5 13 6 15 6 15 7 > 4 q t + 4 q t + 5 q t + 3 q t + 4 q t + 2 q t + 17 7 17 8 19 8 21 9 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n162

K11n161 K11n163