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

K11n53 K11n55

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,6,15,5 X2837 X9,16,10,17 X11,19,12,18 X6,14,7,13 X15,22,16,1 X17,21,18,20 X19,13,20,12 X21,10,22,11

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 4t-2 + 10t-1 - 13 + 10t - 4t2 + t3

Conway Polynomial:

1 + 3z2 + 2z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{10151, K11n129, ...}

Determinant and Signature:

{43, 2}

Jones Polynomial:

- 1 + 3q - 4q2 + 7q3 - 7q4 + 7q5 - 6q6 + 4q7 - 3q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- 1 + q2 - q4 + q6 + 3q8 + q10 + 3q12 - q18 - 3q20 - q24 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 5a-6 - 6a-6z2 - 2a-6z4 + 6a-4 + 10a-4z2 + 5a-4z4 + a-4z6 - a-2 - 2a-2z2 - a-2z4

Kauffman Polynomial:

a-10z2 - 3a-10z4 + a-10z6 - 2a-9z + 9a-9z3 - 11a-9z5 + 3a-9z7 + a-8 - 2a-8z2 + 7a-8z4 - 10a-8z6 + 3a-8z8 - 9a-7z + 23a-7z3 - 18a-7z5 + 2a-7z7 + a-7z9 + 5a-6 - 15a-6z2 + 25a-6z4 - 19a-6z6 + 5a-6z8 - 10a-5z + 19a-5z3 - 9a-5z5 + a-5z9 + 6a-4 - 16a-4z2 + 18a-4z4 - 8a-4z6 + 2a-4z8 - 4a-3z + 6a-3z3 - 2a-3z5 + a-3z7 + a-2 - 4a-2z2 + 3a-2z4 - a-1z + a-1z3

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

{3, 4}

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=2 is the signature of 1154. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                   1 j = 17                 2   j = 15               2 1   j = 13             4 2     j = 11           3 2       j = 9         4 4         j = 7       3 3           j = 5     1 4             j = 3   2 3               j = 1   2                 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, 54]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 54]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], > X[9, 16, 10, 17], X[11, 19, 12, 18], X[6, 14, 7, 13], X[15, 22, 16, 1], > X[17, 21, 18, 20], X[19, 13, 20, 12], X[21, 10, 22, 11]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 54]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, -5, 11, -6, 10, 7, -3, -8, 5, -9, 6, -10, > 9, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 54]]
Out[5]=	DTCode[4, 8, 14, 2, -16, -18, 6, -22, -20, -12, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 54]][t]
Out[6]=	-3 4 10 2 3 -13 + t - -- + -- + 10 t - 4 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 54]][z]
Out[7]=	2 4 6 1 + 3 z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 151], Knot[11, NonAlternating, 54], Knot[11, NonAlternating, 129]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 54]], KnotSignature[Knot[11, NonAlternating, 54]]}
Out[9]=	{43, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 54]][q]
Out[10]=	2 3 4 5 6 7 8 9 -1 + 3 q - 4 q + 7 q - 7 q + 7 q - 6 q + 4 q - 3 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 54]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 54]][q]
Out[12]=	2 4 6 8 10 12 18 20 24 28 -1 + q - q + q + 3 q + q + 3 q - q - 3 q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 54]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -8 5 6 -2 z 6 z 10 z 2 z 2 z 5 z z z a - -- + -- - a + -- - ---- + ----- - ---- - ---- + ---- - -- + -- 6 4 8 6 4 2 6 4 2 4 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 54]][a, z]
Out[14]=	2 2 2 2 -8 5 6 -2 2 z 9 z 10 z 4 z z z 2 z 15 z 16 z a + -- + -- + a - --- - --- - ---- - --- - - + --- - ---- - ----- - ----- - 6 4 9 7 5 3 a 10 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 4 z 9 z 23 z 19 z 6 z z 3 z 7 z 25 z 18 z > ---- + ---- + ----- + ----- + ---- + -- - ---- + ---- + ----- + ----- + 2 9 7 5 3 a 10 8 6 4 a a a a a a a a a 4 5 5 5 5 6 6 6 6 7 3 z 11 z 18 z 9 z 2 z z 10 z 19 z 8 z 3 z > ---- - ----- - ----- - ---- - ---- + --- - ----- - ----- - ---- + ---- + 2 9 7 5 3 10 8 6 4 9 a a a a a a a a a a 7 7 8 8 8 9 9 2 z z 3 z 5 z 2 z z z > ---- + -- + ---- + ---- + ---- + -- + -- 7 3 8 6 4 7 5 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 54]], Vassiliev[3][Knot[11, NonAlternating, 54]]}
Out[15]=	{3, 4}
In[16]:=	Kh[Knot[11, NonAlternating, 54]][q, t]
Out[16]=	3 1 3 5 5 2 7 2 7 3 9 3 2 q + 2 q + --- + 3 q t + q t + 4 q t + 3 q t + 3 q t + 4 q t + q t 9 4 11 4 11 5 13 5 13 6 15 6 15 7 > 4 q t + 3 q t + 2 q t + 4 q t + 2 q t + 2 q t + q t + 17 7 19 8 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n54

K11n53 K11n55