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

K11n11 K11n13

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X7,14,8,15 X2,9,3,10 X11,19,12,18 X13,6,14,7 X15,1,16,22 X17,21,18,20 X19,13,20,12 X21,17,22,16

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

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

Conway Polynomial:

1 + z2 + z4

Other knots with the same Alexander/Conway Polynomial:

{63, ...}

Determinant and Signature:

{13, 0}

Jones Polynomial:

1 - q + 2q2 - 2q3 + 2q4 - 2q5 + 2q6 - q7

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

{943, ...}

A2 (sl(3)) Invariant:

q-2 + 1 + q10 + q14 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 2a-4 + 3a-4z2 + a-4z4 - a-2 - a-2z2 + 1

Kauffman Polynomial:

- 2a-7z + 6a-7z3 - 5a-7z5 + a-7z7 + a-6 - 7a-6z2 + 16a-6z4 - 11a-6z6 + 2a-6z8 - 3a-5z + 7a-5z3 - 4a-5z7 + a-5z9 + 2a-4 - 13a-4z2 + 26a-4z4 - 17a-4z6 + 3a-4z8 - a-3z + 5a-3z5 - 5a-3z7 + a-3z9 + a-2 - 7a-2z2 + 10a-2z4 - 6a-2z6 + a-2z8 - a-1z3 + 1 - z2

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

{1, 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 1112. 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 = 15               1 j = 13             1   j = 11           1 1   j = 9       1 2 1     j = 7       1 1       j = 5     2 2         j = 3 1 1 1           j = 1 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, 12]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 12]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 14, 8, 15], > X[2, 9, 3, 10], X[11, 19, 12, 18], X[13, 6, 14, 7], X[15, 1, 16, 22], > X[17, 21, 18, 20], X[19, 13, 20, 12], X[21, 17, 22, 16]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 12]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, 7, -4, -2, 5, -3, -6, 10, -7, 4, -8, 11, -9, 6, -10, > 9, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 12]]
Out[5]=	DTCode[4, 8, 10, -14, 2, -18, -6, -22, -20, -12, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 12]][t]
Out[6]=	-2 3 2 5 + t - - - 3 t + t t
In[7]:=	Conway[Knot[11, NonAlternating, 12]][z]
Out[7]=	2 4 1 + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[6, 3], Knot[11, NonAlternating, 12]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 12]], KnotSignature[Knot[11, NonAlternating, 12]]}
Out[9]=	{13, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 12]][q]
Out[10]=	2 3 4 5 6 7 1 - q + 2 q - 2 q + 2 q - 2 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 43], Knot[11, NonAlternating, 12]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 12]][q]
Out[12]=	-2 10 14 22 1 + q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 12]][a, z]
Out[13]=	2 2 2 4 -6 2 -2 z 3 z z z 1 - a + -- - a - -- + ---- - -- + -- 4 6 4 2 4 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 12]][a, z]
Out[14]=	2 2 2 3 3 -6 2 -2 2 z 3 z z 2 7 z 13 z 7 z 6 z 7 z 1 + a + -- + a - --- - --- - -- - z - ---- - ----- - ---- + ---- + ---- - 4 7 5 3 6 4 2 7 5 a a a a a a a a a 3 4 4 4 5 5 6 6 6 7 z 16 z 26 z 10 z 5 z 5 z 11 z 17 z 6 z z > -- + ----- + ----- + ----- - ---- + ---- - ----- - ----- - ---- + -- - a 6 4 2 7 3 6 4 2 7 a a a a a a a a a 7 7 8 8 8 9 9 4 z 5 z 2 z 3 z z z z > ---- - ---- + ---- + ---- + -- + -- + -- 5 3 6 4 2 5 3 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 12]], Vassiliev[3][Knot[11, NonAlternating, 12]]}
Out[15]=	{1, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 12]][q, t]
Out[16]=	1 3 3 3 2 5 2 5 3 7 3 9 3 7 4 - + q + q + q t + q t + q t + 2 q t + 2 q t + q t + q t + q t + q 9 4 9 5 11 5 11 6 13 6 15 7 > 2 q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n12

K11n11 K11n13