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

K11n117 K11n119

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X5,15,6,14 X20,8,21,7 X2,10,3,9 X11,17,12,16 X13,18,14,19 X15,9,16,8 X17,1,18,22 X6,20,7,19 X21,12,22,13

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 4t-2 - 4t-1 + 3 - 4t + 4t2 - t3

Conway Polynomial:

1 + 3z2 - 2z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{10160, ...}

Determinant and Signature:

{21, 4}

Jones Polynomial:

2q2 - 2q3 + 3q4 - 4q5 + 4q6 - 3q7 + 2q8 - q9

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

{...}

A2 (sl(3)) Invariant:

2q6 + q8 + 2q10 - 2q18 - q22 + q24 + q26 - q32

HOMFLY-PT Polynomial:

- a-10 + 3a-8 + 4a-8z2 + a-8z4 - 5a-6 - 8a-6z2 - 5a-6z4 - a-6z6 + 4a-4 + 7a-4z2 + 2a-4z4

Kauffman Polynomial:

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

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

{3, 6}

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=4 is the signature of 11118. 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 j = 19               1 j = 17             1   j = 15           2 1   j = 13         2 1     j = 11       2 2       j = 9     1 2         j = 7   1 2           j = 5 1 1             j = 3 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, 118]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 118]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[5, 15, 6, 14], X[20, 8, 21, 7], > X[2, 10, 3, 9], X[11, 17, 12, 16], X[13, 18, 14, 19], X[15, 9, 16, 8], > X[17, 1, 18, 22], X[6, 20, 7, 19], X[21, 12, 22, 13]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 118]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, -10, 4, 8, 5, -2, -6, 11, -7, 3, -8, 6, -9, 7, 10, > -4, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 118]]
Out[5]=	DTCode[4, 10, -14, 20, 2, -16, -18, -8, -22, 6, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 118]][t]
Out[6]=	-3 4 4 2 3 3 - t + -- - - - 4 t + 4 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 118]][z]
Out[7]=	2 4 6 1 + 3 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 160], Knot[11, NonAlternating, 118]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 118]], KnotSignature[Knot[11, NonAlternating, 118]]}
Out[9]=	{21, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 118]][q]
Out[10]=	2 3 4 5 6 7 8 9 2 q - 2 q + 3 q - 4 q + 4 q - 3 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 118]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 118]][q]
Out[12]=	6 8 10 18 22 24 26 32 2 q + q + 2 q - 2 q - q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 118]][a, z]
Out[13]=	2 2 2 4 4 4 6 -10 3 5 4 4 z 8 z 7 z z 5 z 2 z z -a + -- - -- + -- + ---- - ---- + ---- + -- - ---- + ---- - -- 8 6 4 8 6 4 8 6 4 6 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 118]][a, z]
Out[14]=	2 2 2 2 -10 3 5 4 2 z 3 z 3 z 2 z 3 z 9 z 15 z 9 z a + -- + -- + -- - --- - --- - --- - --- - ---- - ---- - ----- - ---- + 8 6 4 11 9 7 5 10 8 6 4 a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 5 z 8 z 10 z 3 z 2 z 12 z 13 z 3 z 4 z 7 z > --- + ---- + ----- + ---- + ---- + ----- + ----- + ---- - ---- - ---- - 11 9 7 5 10 8 6 4 9 7 a a a a a a a a a a 5 6 6 7 7 7 8 8 3 z 5 z 5 z z 2 z z z z > ---- - ---- - ---- + -- + ---- + -- + -- + -- 5 8 6 9 7 5 8 6 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 118]], Vassiliev[3][Knot[11, NonAlternating, 118]]}
Out[15]=	{3, 6}
In[16]:=	Kh[Knot[11, NonAlternating, 118]][q, t]
Out[16]=	3 5 5 7 7 2 9 2 9 3 11 3 11 4 2 q + q + q t + q t + 2 q t + q t + 2 q t + 2 q t + 2 q t + 13 4 13 5 15 5 15 6 17 6 19 7 > 2 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n118

K11n117 K11n119