10 154

10_153

10_155

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10 154 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X8493 X12,6,13,5 X9,17,10,16 X17,1,18,20 X13,19,14,18 X19,15,20,14 X15,11,16,10 X6,12,7,11 X2837
Gauss code	1, -10, 2, -1, 3, -9, 10, -2, -4, 8, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5
Dowker-Thistlethwaite code	4 8 12 2 -16 6 -18 -10 -20 -14
Conway Notation	[(21,2)-(21,2)]

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [5][-15] Hyperbolic Volume 9.24989 A-Polynomial See Data:10 154/A-polynomial

[edit Notes for 10 154's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $3$ Topological 4 genus $[2,3]$ Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for 10 154's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-4t+7-4t^{-1}+t^{-3}$
Conway polynomial	$z^{6}+6z^{4}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 13, 4 }
Jones polynomial	$q^{12}-2q^{11}+2q^{10}-3q^{9}+2q^{8}-2q^{7}+2q^{6}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+6z^{4}a^{-6}+9z^{2}a^{-6}-2z^{2}a^{-8}-2z^{2}a^{-10}+4a^{-6}-2a^{-8}-2a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-10}+z^{8}a^{-12}+z^{7}a^{-9}+3z^{7}a^{-11}+2z^{7}a^{-13}+z^{6}a^{-6}-5z^{6}a^{-10}-3z^{6}a^{-12}+z^{6}a^{-14}-6z^{5}a^{-9}-15z^{5}a^{-11}-9z^{5}a^{-13}-6z^{4}a^{-6}-2z^{4}a^{-8}+7z^{4}a^{-10}-z^{4}a^{-12}-4z^{4}a^{-14}-2z^{3}a^{-7}+9z^{3}a^{-9}+21z^{3}a^{-11}+10z^{3}a^{-13}+9z^{2}a^{-6}+5z^{2}a^{-8}-5z^{2}a^{-10}+2z^{2}a^{-12}+3z^{2}a^{-14}+3za^{-7}-3za^{-9}-10za^{-11}-4za^{-13}-4a^{-6}-2a^{-8}+2a^{-10}+a^{-12}$
The A2 invariant	$q^{-10}+q^{-12}+q^{-14}+2q^{-16}+2q^{-18}+q^{-22}-q^{-24}-q^{-26}-2q^{-28}-2q^{-30}-q^{-34}+q^{-36}+q^{-38}$
The G2 invariant	$q^{-50}+q^{-52}+2q^{-56}+q^{-58}+3q^{-62}+q^{-66}+3q^{-68}-2q^{-70}+3q^{-72}+3q^{-74}-4q^{-76}+7q^{-78}-5q^{-80}+2q^{-82}+5q^{-84}-7q^{-86}+9q^{-88}-4q^{-90}-4q^{-92}+8q^{-94}-5q^{-96}+7q^{-100}-10q^{-102}+8q^{-104}-2q^{-106}-6q^{-108}+8q^{-110}-9q^{-112}+6q^{-114}-5q^{-116}-2q^{-118}+q^{-120}-3q^{-122}-6q^{-126}-2q^{-130}-2q^{-132}+q^{-134}-4q^{-136}-2q^{-138}+8q^{-140}-10q^{-142}+6q^{-144}-7q^{-148}+14q^{-150}-10q^{-152}+6q^{-154}+4q^{-156}-5q^{-158}+8q^{-160}-3q^{-162}-q^{-164}+6q^{-166}-4q^{-168}+3q^{-172}-5q^{-174}+6q^{-176}-4q^{-178}-q^{-180}+q^{-182}-3q^{-184}+2q^{-186}-q^{-188}+q^{-190}$

Further Quantum Invariants

Further quantum knot invariants for 10_154.

A1 Invariants.

Weight	Invariant
1	$q^{-5}+q^{-7}+2q^{-11}-q^{-17}-q^{-19}-q^{-23}+q^{-25}$
2	$q^{-10}+q^{-12}+q^{-14}+3q^{-20}+2q^{-22}-2q^{-24}+q^{-26}+2q^{-28}-2q^{-30}-3q^{-32}-2q^{-38}-2q^{-40}+2q^{-42}-q^{-44}-2q^{-46}+4q^{-48}-2q^{-52}+2q^{-54}+2q^{-56}-q^{-58}-2q^{-60}+q^{-62}+2q^{-64}-2q^{-66}-q^{-68}+q^{-70}$
3	$q^{-15}+q^{-17}+q^{-19}+q^{-21}+3q^{-29}+3q^{-31}+3q^{-33}-2q^{-35}-4q^{-37}-q^{-39}+6q^{-41}+8q^{-43}-6q^{-45}-15q^{-47}-2q^{-49}+14q^{-51}+6q^{-53}-16q^{-55}-12q^{-57}+8q^{-59}+10q^{-61}-2q^{-63}-8q^{-65}+4q^{-69}+3q^{-71}-2q^{-73}-3q^{-75}+q^{-77}+6q^{-79}-6q^{-83}+2q^{-85}+11q^{-87}-12q^{-91}-4q^{-93}+12q^{-95}+8q^{-97}-11q^{-99}-12q^{-101}+2q^{-103}+11q^{-105}+4q^{-107}-6q^{-109}-6q^{-111}+5q^{-115}+4q^{-117}-q^{-119}-4q^{-121}-3q^{-123}+3q^{-125}+3q^{-127}-2q^{-131}-q^{-133}+q^{-135}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-10}+q^{-12}+q^{-14}+2q^{-16}+2q^{-18}+q^{-22}-q^{-24}-q^{-26}-2q^{-28}-2q^{-30}-q^{-34}+q^{-36}+q^{-38}$
1,1	$q^{-20}+2q^{-22}+2q^{-24}+6q^{-26}+5q^{-28}+4q^{-30}+8q^{-32}-4q^{-34}+12q^{-36}-8q^{-38}+2q^{-40}-6q^{-42}-14q^{-44}+8q^{-46}-20q^{-48}+6q^{-50}-8q^{-52}-2q^{-54}+8q^{-56}-8q^{-58}+14q^{-60}-8q^{-62}+12q^{-64}-6q^{-66}+3q^{-68}-8q^{-72}+16q^{-74}-21q^{-76}+22q^{-78}-16q^{-80}+12q^{-82}-2q^{-84}-8q^{-86}+10q^{-88}-12q^{-90}+11q^{-92}-8q^{-94}+4q^{-96}-2q^{-98}+q^{-100}$
2,0	$q^{-20}+q^{-22}+2q^{-24}+q^{-26}+q^{-28}+3q^{-30}+5q^{-32}+2q^{-34}+2q^{-36}+3q^{-38}+3q^{-40}+q^{-42}-2q^{-44}-3q^{-46}-3q^{-48}-4q^{-50}-4q^{-52}-8q^{-54}-6q^{-56}-3q^{-58}-q^{-60}-q^{-62}+3q^{-64}+6q^{-66}+4q^{-68}+4q^{-70}+2q^{-74}+q^{-78}-q^{-80}-q^{-82}+q^{-84}-2q^{-88}-3q^{-90}+q^{-94}+q^{-96}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-20}+q^{-22}+2q^{-24}+3q^{-26}+5q^{-28}+3q^{-30}+5q^{-32}+2q^{-34}-2q^{-40}-3q^{-42}-4q^{-44}-4q^{-46}-4q^{-48}-6q^{-50}-4q^{-52}-q^{-54}+3q^{-58}+6q^{-60}+4q^{-62}+2q^{-64}+2q^{-66}-2q^{-68}-q^{-70}-2q^{-72}-q^{-78}+q^{-80}$
1,0,0	$q^{-15}+q^{-17}+q^{-19}+3q^{-21}+2q^{-23}+2q^{-25}+q^{-27}+q^{-29}-q^{-31}-q^{-33}-2q^{-35}-2q^{-37}-2q^{-39}-2q^{-41}-q^{-45}+q^{-47}+q^{-49}+q^{-51}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-30}+q^{-32}+2q^{-34}+4q^{-36}+6q^{-38}+6q^{-40}+9q^{-42}+7q^{-44}+6q^{-46}+5q^{-48}+3q^{-50}-3q^{-52}-3q^{-54}-3q^{-56}-5q^{-58}-9q^{-60}-9q^{-62}-8q^{-64}-14q^{-66}-13q^{-68}-7q^{-70}-4q^{-72}-q^{-74}+9q^{-76}+12q^{-78}+11q^{-80}+11q^{-82}+10q^{-84}+3q^{-86}-3q^{-88}-3q^{-90}-3q^{-92}-6q^{-94}-4q^{-96}+q^{-104}+q^{-106}$
1,0,0,0	$q^{-20}+q^{-22}+q^{-24}+3q^{-26}+3q^{-28}+2q^{-30}+3q^{-32}+q^{-34}+q^{-36}-q^{-38}-q^{-40}-2q^{-42}-2q^{-44}-2q^{-46}-2q^{-48}-2q^{-50}-2q^{-52}-q^{-56}+q^{-58}+q^{-60}+q^{-62}+q^{-64}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-20}+q^{-22}+3q^{-26}+q^{-28}+q^{-30}+q^{-32}+2q^{-36}-2q^{-38}+2q^{-40}-3q^{-42}+2q^{-44}-2q^{-46}-2q^{-52}+3q^{-54}-4q^{-56}+3q^{-58}-4q^{-60}+2q^{-62}-2q^{-64}-q^{-70}+2q^{-72}-2q^{-74}+2q^{-76}-q^{-78}+q^{-80}$
1,0	$q^{-30}+q^{-34}+q^{-36}+q^{-38}+3q^{-42}+2q^{-44}+2q^{-46}+q^{-48}+2q^{-50}+3q^{-52}+q^{-54}-2q^{-56}+q^{-58}+2q^{-60}-3q^{-64}-2q^{-66}-q^{-70}-4q^{-72}-4q^{-74}-q^{-76}-q^{-78}-2q^{-80}-4q^{-82}-2q^{-84}+q^{-86}+q^{-88}-q^{-90}+2q^{-94}+4q^{-96}+2q^{-98}+q^{-102}+4q^{-104}+q^{-106}-2q^{-108}-2q^{-110}+q^{-112}+q^{-114}-2q^{-116}-2q^{-118}+q^{-120}+q^{-122}-q^{-124}-q^{-126}+q^{-130}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-30}+q^{-32}+q^{-34}+4q^{-36}+3q^{-38}+5q^{-40}+5q^{-42}+5q^{-44}+3q^{-46}+2q^{-48}+2q^{-50}-3q^{-52}-4q^{-56}-5q^{-60}-q^{-62}-4q^{-64}-3q^{-66}-5q^{-68}-6q^{-70}-4q^{-72}-5q^{-74}+q^{-76}-3q^{-78}+5q^{-80}+2q^{-82}+8q^{-84}+2q^{-86}+5q^{-88}+q^{-92}-2q^{-96}-q^{-98}-3q^{-100}+2q^{-102}-2q^{-104}+q^{-106}-q^{-108}+q^{-110}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-50}+q^{-52}+2q^{-56}+q^{-58}+3q^{-62}+q^{-66}+3q^{-68}-2q^{-70}+3q^{-72}+3q^{-74}-4q^{-76}+7q^{-78}-5q^{-80}+2q^{-82}+5q^{-84}-7q^{-86}+9q^{-88}-4q^{-90}-4q^{-92}+8q^{-94}-5q^{-96}+7q^{-100}-10q^{-102}+8q^{-104}-2q^{-106}-6q^{-108}+8q^{-110}-9q^{-112}+6q^{-114}-5q^{-116}-2q^{-118}+q^{-120}-3q^{-122}-6q^{-126}-2q^{-130}-2q^{-132}+q^{-134}-4q^{-136}-2q^{-138}+8q^{-140}-10q^{-142}+6q^{-144}-7q^{-148}+14q^{-150}-10q^{-152}+6q^{-154}+4q^{-156}-5q^{-158}+8q^{-160}-3q^{-162}-q^{-164}+6q^{-166}-4q^{-168}+3q^{-172}-5q^{-174}+6q^{-176}-4q^{-178}-q^{-180}+q^{-182}-3q^{-184}+2q^{-186}-q^{-188}+q^{-190}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 154"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{3}-4t+7-4t^{-1}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+6z^{4}+5z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 13, 4 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{12}-2q^{11}+2q^{10}-3q^{9}+2q^{8}-2q^{7}+2q^{6}+q^{3}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{6}a^{-6}+6z^{4}a^{-6}+9z^{2}a^{-6}-2z^{2}a^{-8}-2z^{2}a^{-10}+4a^{-6}-2a^{-8}-2a^{-10}+a^{-12}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{8}a^{-10}+z^{8}a^{-12}+z^{7}a^{-9}+3z^{7}a^{-11}+2z^{7}a^{-13}+z^{6}a^{-6}-5z^{6}a^{-10}-3z^{6}a^{-12}+z^{6}a^{-14}-6z^{5}a^{-9}-15z^{5}a^{-11}-9z^{5}a^{-13}-6z^{4}a^{-6}-2z^{4}a^{-8}+7z^{4}a^{-10}-z^{4}a^{-12}-4z^{4}a^{-14}-2z^{3}a^{-7}+9z^{3}a^{-9}+21z^{3}a^{-11}+10z^{3}a^{-13}+9z^{2}a^{-6}+5z^{2}a^{-8}-5z^{2}a^{-10}+2z^{2}a^{-12}+3z^{2}a^{-14}+3za^{-7}-3za^{-9}-10za^{-11}-4za^{-13}-4a^{-6}-2a^{-8}+2a^{-10}+a^{-12}$

Vassiliev invariants

V2 and V3:

(5, 9)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $20$ $72$ $200$ ${\frac {1030}{3}}$ ${\frac {146}{3}}$ $1440$ $1840$ $320$ $200$ ${\frac {4000}{3}}$ $2592$ ${\frac {20600}{3}}$ ${\frac {2920}{3}}$ ${\frac {61471}{6}}$ $702$ ${\frac {29318}{9}}$ ${\frac {613}{18}}$ ${\frac {2431}{6}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$4 is the signature of 10 154. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 χ 25                     1 1 23                   1   -1 21                 1 1   0 19               2 1     -1 17           1 1 1       -1 15           2 2         0 13       1 2 1           0 11         2             2 9     1 1               0 7 1                     1 5 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $i=7$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ $r=2$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=9$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=10$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 154]]
Out[2]=	10
In[3]:=	PD[Knot[10, 154]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[9, 17, 10, 16], X[17, 1, 18, 20], X[13, 19, 14, 18], X[19, 15, 20, 14], X[15, 11, 16, 10], X[6, 12, 7, 11], X[2, 8, 3, 7]]
In[4]:=	GaussCode[Knot[10, 154]]
Out[4]=	GaussCode[1, -10, 2, -1, 3, -9, 10, -2, -4, 8, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5]
In[5]:=	BR[Knot[10, 154]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, 1, 3, 2, 2, 2, 3}]
In[6]:=	alex = Alexander[Knot[10, 154]][t]
Out[6]=	-3 4 3 7 + t - - - 4 t + t t
In[7]:=	Conway[Knot[10, 154]][z]
Out[7]=	2 4 6 1 + 5 z + 6 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 154]}
In[9]:=	{KnotDet[Knot[10, 154]], KnotSignature[Knot[10, 154]]}
Out[9]=	{13, 4}
In[10]:=	J=Jones[Knot[10, 154]][q]
Out[10]=	3 6 7 8 9 10 11 12 q + 2 q - 2 q + 2 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, 154]}
In[12]:=	A2Invariant[Knot[10, 154]][q]
Out[12]=	10 12 14 16 18 22 24 26 28 30 q + q + q + 2 q + 2 q + q - q - q - 2 q - 2 q - 34 36 38 q + q + q
In[13]:=	Kauffman[Knot[10, 154]][a, z]
Out[13]=	2 2 2 -12 2 2 4 4 z 10 z 3 z 3 z 3 z 2 z 5 z a + --- - -- - -- - --- - ---- - --- + --- + ---- + ---- - ---- + 10 8 6 13 11 9 7 14 12 10 a a a a a a a a a a 2 2 3 3 3 3 4 4 4 5 z 9 z 10 z 21 z 9 z 2 z 4 z z 7 z ---- + ---- + ----- + ----- + ---- - ---- - ---- - --- + ---- - 8 6 13 11 9 7 14 12 10 a a a a a a a a a 4 4 5 5 5 6 6 6 6 7 2 z 6 z 9 z 15 z 6 z z 3 z 5 z z 2 z ---- - ---- - ---- - ----- - ---- + --- - ---- - ---- + -- + ---- + 8 6 13 11 9 14 12 10 6 13 a a a a a a a a a a 7 7 8 8 3 z z z z ---- + -- + --- + --- 11 9 12 10 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 154]], Vassiliev[3][Knot[10, 154]]}
Out[14]=	{0, 9}
In[15]:=	Kh[Knot[10, 154]][q, t]
Out[15]=	5 7 9 2 9 3 13 3 11 4 13 4 13 5 q + q + q t + q t + q t + 2 q t + 2 q t + q t + 15 5 17 5 15 6 17 6 17 7 19 7 19 8 2 q t + q t + 2 q t + q t + q t + 2 q t + q t + 21 8 21 9 23 9 25 10 q t + q t + q t + q t