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

K11n17 K11n19

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X7,16,8,17 X2,9,3,10 X11,19,12,18 X13,21,14,20 X15,6,16,7 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-2 - 8t-1 + 13 - 8t + 2t2

Conway Polynomial:

1 + 2z4

Other knots with the same Alexander/Conway Polynomial:

{10146, K11n62, ...}

Determinant and Signature:

{33, 0}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

q-8 + q-6 - q-4 + q-2 - q2 - q6 + q8 + q10 + q12 + 2q14 - q16 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 2a-4 + 2a-4z2 + a-4z4 + a-2z2 + a-2z4 - 1 - 2z2 + a2

Kauffman Polynomial:

- 2a-7z + 7a-7z3 - 5a-7z5 + a-7z7 + a-6 - 8a-6z2 + 15a-6z4 - 10a-6z6 + 2a-6z8 - 3a-5z + 9a-5z3 - 5a-5z5 - 2a-5z7 + a-5z9 + 2a-4 - 12a-4z2 + 21a-4z4 - 17a-4z6 + 4a-4z8 - a-3z + 6a-3z3 - 6a-3z5 - a-3z7 + a-3z9 - a-2z2 + 4a-2z4 - 6a-2z6 + 2a-2z8 - a-1z + 6a-1z3 - 6a-1z5 + 2a-1z7 - 1 + 4z2 - 2z4 + z6 - az + 2az3 - a2 + a2z2

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

{0, 2}

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=0 is the signature of 1118. 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 = 15                   1 j = 13                 1   j = 11               2 1   j = 9             3 1     j = 7           2 2       j = 5         3 3         j = 3       2 2           j = 1     2 3             j = -1   1 3               j = -3   1                 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, 18]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 18]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 16, 8, 17], > X[2, 9, 3, 10], X[11, 19, 12, 18], X[13, 21, 14, 20], X[15, 6, 16, 7], > X[17, 1, 18, 22], X[19, 15, 20, 14], X[21, 13, 22, 12]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 18]]
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, 18]]
Out[5]=	DTCode[4, 8, 10, -16, 2, -18, -20, -6, -22, -14, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 18]][t]
Out[6]=	2 8 2 13 + -- - - - 8 t + 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 18]][z]
Out[7]=	4 1 + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 146], Knot[11, NonAlternating, 18], Knot[11, NonAlternating, 62]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 18]], KnotSignature[Knot[11, NonAlternating, 18]]}
Out[9]=	{33, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 18]][q]
Out[10]=	-2 2 2 3 4 5 6 7 4 + q - - - 5 q + 5 q - 5 q + 5 q - 3 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 18]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 18]][q]
Out[12]=	-8 -6 -4 -2 2 6 8 10 12 14 16 22 q + q - q + q - q - q + q + q + q + 2 q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 18]][a, z]
Out[13]=	2 2 2 4 4 -6 2 2 2 z 2 z z z z -1 - a + -- + a - 2 z - -- + ---- + -- + -- + -- 4 6 4 2 4 2 a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 18]][a, z]
Out[14]=	2 2 2 -6 2 2 2 z 3 z z z 2 8 z 12 z z -1 + a + -- - a - --- - --- - -- - - - a z + 4 z - ---- - ----- - -- + 4 7 5 3 a 6 4 2 a a a a a a a 3 3 3 3 4 4 4 2 2 7 z 9 z 6 z 6 z 3 4 15 z 21 z 4 z > a z + ---- + ---- + ---- + ---- + 2 a z - 2 z + ----- + ----- + ---- - 7 5 3 a 6 4 2 a a a a a a 5 5 5 5 6 6 6 7 7 7 5 z 5 z 6 z 6 z 6 10 z 17 z 6 z z 2 z z > ---- - ---- - ---- - ---- + z - ----- - ----- - ---- + -- - ---- - -- + 7 5 3 a 6 4 2 7 5 3 a a a a a a a a a 7 8 8 8 9 9 2 z 2 z 4 z 2 z z z > ---- + ---- + ---- + ---- + -- + -- a 6 4 2 5 3 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 18]], Vassiliev[3][Knot[11, NonAlternating, 18]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 18]][q, t]
Out[16]=	3 1 1 1 3 3 2 5 2 5 3 - + 2 q + ----- + ---- + --- + 3 q t + 2 q t + 2 q t + 3 q t + 3 q t + q 5 2 3 q t q t q t 7 3 7 4 9 4 9 5 11 5 11 6 13 6 15 7 > 2 q t + 2 q t + 3 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n18

K11n17 K11n19