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

K11n26 K11n28

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,13,6,12 X2837 X9,15,10,14 X11,18,12,19 X13,7,14,6 X15,21,16,20 X17,1,18,22 X19,10,20,11 X21,17,22,16

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-4 + 3t-3 - 3t-2 + 2t-1 - 1 + 2t - 3t2 + 3t3 - t4

Conway Polynomial:

1 + z2 - 5z4 - 5z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{19, 6}

Jones Polynomial:

q - q2 + 3q3 - 3q4 + 3q5 - 3q6 + 2q7 - 2q8 + q9

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

{10133, ...}

A2 (sl(3)) Invariant:

q4 + q6 + 2q8 + 2q10 + q12 - 2q16 - q18 - 3q20 + q26 + q28 + q32 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 5a-8 + 10a-8z2 + 6a-8z4 + a-8z6 - 9a-6 - 19a-6z2 - 17a-6z4 - 7a-6z6 - a-6z8 + 6a-4 + 11a-4z2 + 6a-4z4 + a-4z6

Kauffman Polynomial:

- a-11z + a-10 + a-10z2 - 4a-10z4 + a-10z6 - 6a-9z + 18a-9z3 - 15a-9z5 + 3a-9z7 + 5a-8 - 15a-8z2 + 24a-8z4 - 16a-8z6 + 3a-8z8 - 10a-7z + 22a-7z3 - 10a-7z5 - 2a-7z7 + a-7z9 + 9a-6 - 33a-6z2 + 45a-6z4 - 24a-6z6 + 4a-6z8 - 5a-5z + 4a-5z3 + 5a-5z5 - 5a-5z7 + a-5z9 + 6a-4 - 17a-4z2 + 17a-4z4 - 7a-4z6 + a-4z8

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

{1, 0}

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=6 is the signature of 1127. 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 = 19                 1 j = 17               1   j = 15             2 2   j = 13           2 1     j = 11         1 2 1     j = 9       2 2         j = 7     1 1           j = 5   1 3             j = 3                   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, 27]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 27]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 13, 6, 12], X[2, 8, 3, 7], > X[9, 15, 10, 14], X[11, 18, 12, 19], X[13, 7, 14, 6], X[15, 21, 16, 20], > X[17, 1, 18, 22], X[19, 10, 20, 11], X[21, 17, 22, 16]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 27]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 3, -7, 5, -8, 11, -9, 6, -10, > 8, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 27]]
Out[5]=	DTCode[4, 8, -12, 2, -14, -18, -6, -20, -22, -10, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 27]][t]
Out[6]=	-4 3 3 2 2 3 4 -1 - t + -- - -- + - + 2 t - 3 t + 3 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 27]][z]
Out[7]=	2 4 6 8 1 + z - 5 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 27]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 27]], KnotSignature[Knot[11, NonAlternating, 27]]}
Out[9]=	{19, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 27]][q]
Out[10]=	2 3 4 5 6 7 8 9 q - q + 3 q - 3 q + 3 q - 3 q + 2 q - 2 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 133], Knot[11, NonAlternating, 27]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 27]][q]
Out[12]=	4 6 8 10 12 16 18 20 26 28 32 34 q + q + 2 q + 2 q + q - 2 q - q - 3 q + q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 27]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -10 5 9 6 z 10 z 19 z 11 z 6 z 17 z 6 z z -a + -- - -- + -- - --- + ----- - ----- + ----- + ---- - ----- + ---- + -- - 8 6 4 10 8 6 4 8 6 4 8 a a a a a a a a a a a 6 6 8 7 z z z > ---- + -- - -- 6 4 6 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 27]][a, z]
Out[14]=	2 2 2 2 -10 5 9 6 z 6 z 10 z 5 z z 15 z 33 z 17 z a + -- + -- + -- - --- - --- - ---- - --- + --- - ----- - ----- - ----- + 8 6 4 11 9 7 5 10 8 6 4 a a a a a a a a a a a 3 3 3 4 4 4 4 5 5 18 z 22 z 4 z 4 z 24 z 45 z 17 z 15 z 10 z > ----- + ----- + ---- - ---- + ----- + ----- + ----- - ----- - ----- + 9 7 5 10 8 6 4 9 7 a a a a a a a a a 5 6 6 6 6 7 7 7 8 8 8 5 z z 16 z 24 z 7 z 3 z 2 z 5 z 3 z 4 z z > ---- + --- - ----- - ----- - ---- + ---- - ---- - ---- + ---- + ---- + -- + 5 10 8 6 4 9 7 5 8 6 4 a a a a a a a a a a a 9 9 z z > -- + -- 7 5 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 27]], Vassiliev[3][Knot[11, NonAlternating, 27]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 27]][q, t]
Out[16]=	5 5 7 q q 7 9 9 2 11 2 11 3 13 3 3 q + q + -- + -- + q t + 2 q t + 2 q t + q t + 2 q t + 2 q t + 2 t t 11 4 13 4 15 4 15 5 17 5 19 6 > q t + q t + 2 q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n27

K11n26 K11n28