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

K11n15 K11n17

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X16,8,17,7 X2,9,3,10 X11,19,12,18 X13,21,14,20 X6,16,7,15 X17,1,18,22 X19,15,20,14 X21,13,22,12

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-3 + 7t-2 - 7t-1 + 5 - 7t + 7t2 - 2t3

Conway Polynomial:

1 + 3z2 - 5z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{37, 4}

Jones Polynomial:

1 - 2q + 4q2 - 5q3 + 6q4 - 6q5 + 6q6 - 4q7 + 2q8 - q9

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

{936, ...}

A2 (sl(3)) Invariant:

1 + q6 + 2q10 - q12 + q14 - q18 + q20 - q22 + q24 - q32

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-11z + a-11z3 + a-10 - 3a-10z2 + 2a-10z4 - 2a-9z + 6a-9z3 - 3a-9z5 + a-9z7 + 2a-8 - 9a-8z2 + 17a-8z4 - 9a-8z6 + 2a-8z8 - 2a-7z + 6a-7z3 - a-7z5 - 2a-7z7 + a-7z9 + 2a-6 - 12a-6z2 + 22a-6z4 - 16a-6z6 + 4a-6z8 - 3a-5z + 6a-5z3 - 5a-5z5 - a-5z7 + a-5z9 + a-4 - 2a-4z2 + 3a-4z4 - 6a-4z6 + 2a-4z8 - a-3z + 5a-3z3 - 7a-3z5 + 2a-3z7 - a-2 + 4a-2z2 - 4a-2z4 + a-2z6

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

{3, 7}

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 1116. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                   1 j = 17                 1   j = 15               3 1   j = 13             3 1     j = 11           3 3       j = 9         3 3         j = 7       2 3           j = 5     2 3             j = 3   1 3               j = 1   1                 j = -1 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, 16]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 16]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[16, 8, 17, 7], > X[2, 9, 3, 10], X[11, 19, 12, 18], X[13, 21, 14, 20], X[6, 16, 7, 15], > X[17, 1, 18, 22], X[19, 15, 20, 14], X[21, 13, 22, 12]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 16]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -3, -6, 11, -7, 10, 8, -4, -9, 6, -10, > 7, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 16]]
Out[5]=	DTCode[4, 8, 10, 16, 2, -18, -20, 6, -22, -14, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 16]][t]
Out[6]=	2 7 7 2 3 5 - -- + -- - - - 7 t + 7 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 16]][z]
Out[7]=	2 4 6 1 + 3 z - 5 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 16]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 16]], KnotSignature[Knot[11, NonAlternating, 16]]}
Out[9]=	{37, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 16]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 2 q + 4 q - 5 q + 6 q - 6 q + 6 q - 4 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 36], Knot[11, NonAlternating, 16]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 16]][q]
Out[12]=	6 10 12 14 18 20 22 24 32 1 + q + 2 q - q + q - q + q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 16]][a, z]
Out[13]=	2 2 2 4 4 4 4 6 -10 2 2 -4 -2 4 z 4 z 3 z z 4 z 3 z z z -a + -- - -- + a + a + ---- - ---- + ---- + -- - ---- - ---- + -- - -- - 8 6 8 6 2 8 6 4 2 6 a a a a a a a a a a 6 z > -- 4 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 16]][a, z]
Out[14]=	2 2 2 -10 2 2 -4 -2 2 z 2 z 2 z 3 z z 3 z 9 z 12 z a + -- + -- + a - a - --- - --- - --- - --- - -- - ---- - ---- - ----- - 8 6 11 9 7 5 3 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 4 z z 6 z 6 z 6 z 5 z 2 z 17 z 22 z > ---- + ---- + --- + ---- + ---- + ---- + ---- + ---- + ----- + ----- + 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a 4 4 5 5 5 5 6 6 6 6 7 3 z 4 z 3 z z 5 z 7 z 9 z 16 z 6 z z z > ---- - ---- - ---- - -- - ---- - ---- - ---- - ----- - ---- + -- + -- - 4 2 9 7 5 3 8 6 4 2 9 a a a a a a a a a a a 7 7 7 8 8 8 9 9 2 z z 2 z 2 z 4 z 2 z z z > ---- - -- + ---- + ---- + ---- + ---- + -- + -- 7 5 3 8 6 4 7 5 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 16]], Vassiliev[3][Knot[11, NonAlternating, 16]]}
Out[15]=	{3, 7}
In[16]:=	Kh[Knot[11, NonAlternating, 16]][q, t]
Out[16]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 3 q + 2 q + ---- + - + -- + 3 q t + 2 q t + 3 q t + 3 q t + 3 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 17 6 > 3 q t + 3 q t + 3 q t + q t + 3 q t + q t + q t + 19 7 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n16

K11n15 K11n17