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

K11a27 K11a29

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X12,6,13,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,14,21,13 X22,16,1,15 X10,17,11,18 X6,19,7,20 X14,22,15,21

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-4 - 6t-3 + 15t-2 - 24t-1 + 29 - 24t + 15t2 - 6t3 + t4

Conway Polynomial:

1 - 2z2 - z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{10123, ...}

Determinant and Signature:

{121, 0}

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 12q-3 + 16q-2 - 19q-1 + 20 - 17q + 13q2 - 8q3 + 4q4 - q5

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

{K11a87, K11a96, ...}

A2 (sl(3)) Invariant:

q-18 + 2q-12 - 3q-10 + 2q-8 - 2q-6 - 2q-4 + 3q-2 - 3 + 5q2 - 2q4 + q6 + 2q8 - 2q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- 2a-2z2 - 3a-2z4 - a-2z6 + 3 + 7z2 + 9z4 + 5z6 + z8 - 4a2 - 10a2z2 - 8a2z4 - 2a2z6 + 2a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + a-4z2 - 6a-4z4 + 4a-4z6 + 3a-3z3 - 11a-3z5 + 7a-3z7 - 2a-2z2 + 7a-2z4 - 13a-2z6 + 8a-2z8 + a-1z + 2a-1z5 - 7a-1z7 + 6a-1z9 + 3 - 16z2 + 37z4 - 31z6 + 9z8 + 2z10 + 3az - 15az3 + 31az5 - 28az7 + 11az9 + 4a2 - 22a2z2 + 38a2z4 - 28a2z6 + 6a2z8 + 2a2z10 + a3z - 6a3z3 + 9a3z5 - 11a3z7 + 5a3z9 + 2a4 - 7a4z2 + 11a4z4 - 13a4z6 + 5a4z8 - a5z + 5a5z3 - 8a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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

{-2, 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 1128. 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 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     3   j = 7                   5 1   j = 5                 8 3     j = 3               9 5       j = 1             11 8         j = -1           9 10           j = -3         7 10             j = -5       5 9               j = -7     2 7                 j = -9   1 5                   j = -11   2                     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, Alternating, 28]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 28]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[12, 6, 13, 5], X[16, 7, 17, 8], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[20, 14, 21, 13], X[22, 16, 1, 15], > X[10, 17, 11, 18], X[6, 19, 7, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 28]]
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, Alternating, 28]]
Out[5]=	DTCode[4, 8, 12, 16, 2, 18, 20, 22, 10, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 28]][t]
Out[6]=	-4 6 15 24 2 3 4 29 + t - -- + -- - -- - 24 t + 15 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 28]][z]
Out[7]=	2 4 6 8 1 - 2 z - z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 123], Knot[11, Alternating, 28]}
In[9]:=	{KnotDet[Knot[11, Alternating, 28]], KnotSignature[Knot[11, Alternating, 28]]}
Out[9]=	{121, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 28]][q]
Out[10]=	-6 3 7 12 16 19 2 3 4 5 20 + q - -- + -- - -- + -- - -- - 17 q + 13 q - 8 q + 4 q - 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, Alternating, 28], Knot[11, Alternating, 87], > Knot[11, Alternating, 96]}
In[12]:=	A2Invariant[Knot[11, Alternating, 28]][q]
Out[12]=	-18 2 3 2 2 2 3 2 4 6 8 10 -3 + q + --- - --- + -- - -- - -- + -- + 5 q - 2 q + q + 2 q - 2 q + 12 10 8 6 4 2 q q q q q q 12 14 > 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 28]][a, z]
Out[13]=	2 4 2 4 2 2 z 2 2 4 2 4 3 z 2 4 3 - 4 a + 2 a + 7 z - ---- - 10 a z + 3 a z + 9 z - ---- - 8 a z + 2 2 a a 6 4 4 6 z 2 6 8 > a z + 5 z - -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 28]][a, z]
Out[14]=	2 2 2 4 z 3 5 2 z 2 z 2 2 3 + 4 a + 2 a + - + 3 a z + a z - a z - 16 z + -- - ---- - 22 a z - a 4 2 a a 3 3 4 2 6 2 z 3 z 3 3 3 5 3 4 > 7 a z + 2 a z - -- + ---- - 15 a z - 6 a z + 5 a z + 37 z - 5 3 a a 4 4 5 5 5 6 z 7 z 2 4 4 4 6 4 z 11 z 2 z 5 > ---- + ---- + 38 a z + 11 a z - 3 a z + -- - ----- + ---- + 31 a z + 4 2 5 3 a a a a a 6 6 3 5 5 5 6 4 z 13 z 2 6 4 6 6 6 > 9 a z - 8 a z - 31 z + ---- - ----- - 28 a z - 13 a z + a z + 4 2 a a 7 7 8 7 z 7 z 7 3 7 5 7 8 8 z 2 8 > ---- - ---- - 28 a z - 11 a z + 3 a z + 9 z + ---- + 6 a z + 3 a 2 a a 9 4 8 6 z 9 3 9 10 2 10 > 5 a z + ---- + 11 a z + 5 a z + 2 z + 2 a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 28]], Vassiliev[3][Knot[11, Alternating, 28]]}
Out[15]=	{-2, 2}
In[16]:=	Kh[Knot[11, Alternating, 28]][q, t]
Out[16]=	10 1 2 1 5 2 7 5 9 -- + 11 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 7 10 9 3 3 2 5 2 5 3 > ----- + ---- + --- + 8 q t + 9 q t + 5 q t + 8 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a28

K11a27 K11a29