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

K11n47 K11n49

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X5,12,6,13 X7,17,8,16 X2,9,3,10 X11,19,12,18 X13,20,14,21 X15,22,16,1 X17,11,18,10 X19,7,20,6 X21,14,22,15

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 3t-2 - 6t-1 + 9 - 6t + 3t2 - t3

Conway Polynomial:

1 - 3z2 - 3z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{29, 0}

Jones Polynomial:

q-6 - 2q-5 + 3q-4 - 4q-3 + 5q-2 - 5q-1 + 4 - 3q + 2q2

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

{...}

A2 (sl(3)) Invariant:

q-18 + q-14 - q-10 + q-8 - q-6 - q-2 - 1 + q2 + 2q6 + q8

HOMFLY-PT Polynomial:

a-2 + 1 + 2z2 + z4 - 3a2 - 8a2z2 - 5a2z4 - a2z6 + 2a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- a-2 + 2a-2z2 + a-1z5 + 1 - 6z2 + 11z4 - 5z6 + z8 + 2az - 9az3 + 11az5 - 5az7 + az9 + 3a2 - 17a2z2 + 24a2z4 - 14a2z6 + 3a2z8 + a3z - 2a3z3 + 2a3z5 - 3a3z7 + a3z9 + 2a4 - 6a4z2 + 9a4z4 - 8a4z6 + 2a4z8 - a5z + 7a5z3 - 8a5z5 + 2a5z7 + 3a6z2 - 4a6z4 + a6z6

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

{-3, 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 1148. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                 2 j = 3               1   j = 1             3 2   j = -1           3 2     j = -3         2 2       j = -5       2 3         j = -7     1 2           j = -9   1 2             j = -11   1               j = -13 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, 48]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 48]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[5, 12, 6, 13], X[7, 17, 8, 16], > X[2, 9, 3, 10], X[11, 19, 12, 18], X[13, 20, 14, 21], X[15, 22, 16, 1], > X[17, 11, 18, 10], X[19, 7, 20, 6], X[21, 14, 22, 15]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 48]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 3, -7, 11, -8, 4, -9, 6, -10, > 7, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 48]]
Out[5]=	DTCode[4, 8, -12, -16, 2, -18, -20, -22, -10, -6, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 48]][t]
Out[6]=	-3 3 6 2 3 9 - t + -- - - - 6 t + 3 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 48]][z]
Out[7]=	2 4 6 1 - 3 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 48]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 48]], KnotSignature[Knot[11, NonAlternating, 48]]}
Out[9]=	{29, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 48]][q]
Out[10]=	-6 2 3 4 5 5 2 4 + q - -- + -- - -- + -- - - - 3 q + 2 q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 48]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 48]][q]
Out[12]=	-18 -14 -10 -8 -6 -2 2 6 8 -1 + q + q - q + q - q - q + q + 2 q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 48]][a, z]
Out[13]=	-2 2 4 2 2 2 4 2 4 2 4 4 4 2 6 1 + a - 3 a + 2 a + 2 z - 8 a z + 3 a z + z - 5 a z + a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 48]][a, z]
Out[14]=	2 -2 2 4 3 5 2 2 z 2 2 1 - a + 3 a + 2 a + 2 a z + a z - a z - 6 z + ---- - 17 a z - 2 a 4 2 6 2 3 3 3 5 3 4 2 4 > 6 a z + 3 a z - 9 a z - 2 a z + 7 a z + 11 z + 24 a z + 5 4 4 6 4 z 5 3 5 5 5 6 2 6 > 9 a z - 4 a z + -- + 11 a z + 2 a z - 8 a z - 5 z - 14 a z - a 4 6 6 6 7 3 7 5 7 8 2 8 4 8 > 8 a z + a z - 5 a z - 3 a z + 2 a z + z + 3 a z + 2 a z + 9 3 9 > a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 48]], Vassiliev[3][Knot[11, NonAlternating, 48]]}
Out[15]=	{-3, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 48]][q, t]
Out[16]=	2 1 1 1 2 1 2 2 3 - + 3 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 2 2 3 3 5 2 > ----- + ---- + --- + 2 q t + q t + 2 q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n48

K11n47 K11n49