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

K11n109 K11n111

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X14,6,15,5 X7,17,8,16 X9,19,10,18 X2,11,3,12 X20,13,21,14 X22,16,1,15 X17,9,18,8 X12,19,13,20 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 4t-2 - 9t-1 + 13 - 9t + 4t2 - t3

Conway Polynomial:

1 - 2z2 - 2z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{41, 0}

Jones Polynomial:

2q-2 - 4q-1 + 6 - 7q + 7q2 - 6q3 + 5q4 - 3q5 + q6

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

{...}

A2 (sl(3)) Invariant:

q-8 + 2q-6 - q-4 + q-2 - 1 - q2 + q4 - q6 + 2q8 - q10 + q14 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 - a-2 - 5a-2z2 - 4a-2z4 - a-2z6 + z2 + z4 + a2

Kauffman Polynomial:

2a-6z2 - 3a-6z4 + a-6z6 - a-5z + 7a-5z3 - 10a-5z5 + 3a-5z7 + a-4 - 2a-4z2 + 5a-4z4 - 9a-4z6 + 3a-4z8 - a-3z + 7a-3z3 - 10a-3z5 + a-3z7 + a-3z9 + a-2 - 7a-2z2 + 14a-2z4 - 13a-2z6 + 4a-2z8 - a-1z + 2a-1z3 + a-1z5 - 2a-1z7 + a-1z9 - z2 + 6z4 - 3z6 + z8 - az + 2az3 + az5 - a2 + 2a2z2

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=0 is the signature of 11110. 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 j = 13                 1 j = 11               2   j = 9             3 1   j = 7           3 2     j = 5         4 3       j = 3       3 3         j = 1     3 4           j = -1   2 4             j = -3   2               j = -5 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, 110]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 110]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 6, 15, 5], X[7, 17, 8, 16], > X[9, 19, 10, 18], X[2, 11, 3, 12], X[20, 13, 21, 14], X[22, 16, 1, 15], > X[17, 9, 18, 8], X[12, 19, 13, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 110]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, -4, 9, -5, -2, 6, -10, 7, -3, 8, 4, -9, 5, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 110]]
Out[5]=	DTCode[4, 10, 14, -16, -18, 2, 20, 22, -8, 12, 6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 110]][t]
Out[6]=	-3 4 9 2 3 13 - t + -- - - - 9 t + 4 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 110]][z]
Out[7]=	2 4 6 1 - 2 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 110]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 110]], KnotSignature[Knot[11, NonAlternating, 110]]}
Out[9]=	{41, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 110]][q]
Out[10]=	2 4 2 3 4 5 6 6 + -- - - - 7 q + 7 q - 6 q + 5 q - 3 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 110]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 110]][q]
Out[12]=	-8 2 -4 -2 2 4 6 8 10 14 16 18 -1 + q + -- - q + q - q + q - q + 2 q - q + q - q + q 6 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 110]][a, z]
Out[13]=	2 2 4 4 6 -4 -2 2 2 2 z 5 z 4 z 4 z z a - a + a + z + ---- - ---- + z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 110]][a, z]
Out[14]=	2 2 2 3 -4 -2 2 z z z 2 2 z 2 z 7 z 2 2 7 z a + a - a - -- - -- - - - a z - z + ---- - ---- - ---- + 2 a z + ---- + 5 3 a 6 4 2 5 a a a a a a 3 3 4 4 4 5 5 5 7 z 2 z 3 4 3 z 5 z 14 z 10 z 10 z z > ---- + ---- + 2 a z + 6 z - ---- + ---- + ----- - ----- - ----- + -- + 3 a 6 4 2 5 3 a a a a a a a 6 6 6 7 7 7 8 8 5 6 z 9 z 13 z 3 z z 2 z 8 3 z 4 z > a z - 3 z + -- - ---- - ----- + ---- + -- - ---- + z + ---- + ---- + 6 4 2 5 3 a 4 2 a a a a a a a 9 9 z z > -- + -- 3 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 110]], Vassiliev[3][Knot[11, NonAlternating, 110]]}
Out[15]=	{-2, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 110]][q, t]
Out[16]=	4 2 2 2 3 3 2 5 2 5 3 - + 3 q + ----- + ---- + --- + 4 q t + 3 q t + 3 q t + 4 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 13 6 > 3 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n110

K11n109 K11n111