10 101

10_100

10_102

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10 101 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X16,12,17,11 X12,20,13,19 X18,8,19,7 X6,14,7,13 X8,18,9,17 X2,10,3,9
Gauss code	1, -10, 2, -1, 3, -8, 7, -9, 10, -2, 5, -6, 8, -3, 4, -5, 9, -7, 6, -4
Dowker-Thistlethwaite code	4 10 14 18 2 16 6 20 8 12
Conway Notation	[21:2:2]

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [3][-15] Hyperbolic Volume 14.6875 A-Polynomial See Data:10 101/A-polynomial

[edit Notes for 10 101's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $2$ Rasmussen s-Invariant -4

[edit Notes for 10 101's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$7t^{2}-21t+29-21t^{-1}+7t^{-2}$
Conway polynomial	$7z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 85, 4 }
Jones polynomial	$q^{12}-4q^{11}+7q^{10}-11q^{9}+13q^{8}-14q^{7}+14q^{6}-10q^{5}+7q^{4}-3q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+3z^{4}a^{-6}+3z^{4}a^{-8}+z^{2}a^{-4}+5z^{2}a^{-6}+5z^{2}a^{-8}-4z^{2}a^{-10}+2a^{-6}+2a^{-8}-4a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-9}+2z^{9}a^{-11}+6z^{8}a^{-8}+11z^{8}a^{-10}+5z^{8}a^{-12}+7z^{7}a^{-7}+10z^{7}a^{-9}+7z^{7}a^{-11}+4z^{7}a^{-13}+6z^{6}a^{-6}-6z^{6}a^{-8}-24z^{6}a^{-10}-11z^{6}a^{-12}+z^{6}a^{-14}+3z^{5}a^{-5}-8z^{5}a^{-7}-31z^{5}a^{-9}-31z^{5}a^{-11}-11z^{5}a^{-13}+z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}+15z^{4}a^{-10}+3z^{4}a^{-12}-2z^{4}a^{-14}-2z^{3}a^{-5}+4z^{3}a^{-7}+26z^{3}a^{-9}+28z^{3}a^{-11}+8z^{3}a^{-13}-z^{2}a^{-4}+7z^{2}a^{-6}-z^{2}a^{-8}-9z^{2}a^{-10}+z^{2}a^{-12}+z^{2}a^{-14}-8za^{-9}-9za^{-11}-za^{-13}-2a^{-6}+2a^{-8}+4a^{-10}+a^{-12}$
The A2 invariant	$q^{-6}-2q^{-8}+2q^{-10}+q^{-12}-2q^{-14}+4q^{-16}+2q^{-20}+2q^{-22}-q^{-24}+2q^{-26}-4q^{-28}-q^{-30}-3q^{-34}+q^{-36}+q^{-38}$
The G2 invariant	$q^{-30}-2q^{-32}+4q^{-34}-6q^{-36}+6q^{-38}-5q^{-40}+11q^{-44}-21q^{-46}+33q^{-48}-38q^{-50}+31q^{-52}-14q^{-54}-15q^{-56}+52q^{-58}-84q^{-60}+107q^{-62}-105q^{-64}+68q^{-66}+3q^{-68}-87q^{-70}+169q^{-72}-206q^{-74}+183q^{-76}-94q^{-78}-40q^{-80}+166q^{-82}-226q^{-84}+203q^{-86}-85q^{-88}-58q^{-90}+169q^{-92}-188q^{-94}+107q^{-96}+45q^{-98}-192q^{-100}+265q^{-102}-220q^{-104}+73q^{-106}+125q^{-108}-289q^{-110}+359q^{-112}-307q^{-114}+147q^{-116}+50q^{-118}-229q^{-120}+322q^{-122}-304q^{-124}+183q^{-126}-14q^{-128}-146q^{-130}+224q^{-132}-204q^{-134}+85q^{-136}+65q^{-138}-188q^{-140}+214q^{-142}-138q^{-144}-16q^{-146}+177q^{-148}-270q^{-150}+259q^{-152}-149q^{-154}-14q^{-156}+158q^{-158}-232q^{-160}+224q^{-162}-139q^{-164}+32q^{-166}+59q^{-168}-106q^{-170}+105q^{-172}-71q^{-174}+33q^{-176}+q^{-178}-19q^{-180}+20q^{-182}-16q^{-184}+8q^{-186}-3q^{-188}+q^{-190}$

Further Quantum Invariants

Further quantum knot invariants for 10_101.

A1 Invariants.

Weight	Invariant
1	$q^{-3}-2q^{-5}+4q^{-7}-3q^{-9}+4q^{-11}-q^{-15}+2q^{-17}-4q^{-19}+3q^{-21}-3q^{-23}+q^{-25}$
2	$q^{-6}-2q^{-8}+q^{-10}+6q^{-12}-9q^{-14}+3q^{-16}+17q^{-18}-24q^{-20}+35q^{-24}-26q^{-26}-15q^{-28}+34q^{-30}-7q^{-32}-22q^{-34}+11q^{-36}+16q^{-38}-16q^{-40}-15q^{-42}+27q^{-44}-2q^{-46}-33q^{-48}+25q^{-50}+16q^{-52}-35q^{-54}+9q^{-56}+23q^{-58}-18q^{-60}-5q^{-62}+12q^{-64}-2q^{-66}-3q^{-68}+q^{-70}$
3	$q^{-9}-2q^{-11}+q^{-13}+3q^{-15}-5q^{-19}+3q^{-21}+11q^{-23}-11q^{-25}-19q^{-27}+30q^{-29}+42q^{-31}-44q^{-33}-90q^{-35}+59q^{-37}+150q^{-39}-41q^{-41}-214q^{-43}-7q^{-45}+255q^{-47}+78q^{-49}-255q^{-51}-147q^{-53}+201q^{-55}+200q^{-57}-121q^{-59}-219q^{-61}+32q^{-63}+201q^{-65}+58q^{-67}-176q^{-69}-125q^{-71}+130q^{-73}+181q^{-75}-95q^{-77}-218q^{-79}+44q^{-81}+250q^{-83}+12q^{-85}-260q^{-87}-79q^{-89}+248q^{-91}+146q^{-93}-198q^{-95}-201q^{-97}+124q^{-99}+226q^{-101}-41q^{-103}-202q^{-105}-36q^{-107}+151q^{-109}+78q^{-111}-86q^{-113}-82q^{-115}+29q^{-117}+59q^{-119}+4q^{-121}-34q^{-123}-9q^{-125}+11q^{-127}+7q^{-129}-2q^{-131}-3q^{-133}+q^{-135}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-6}-2q^{-8}+2q^{-10}+q^{-12}-2q^{-14}+4q^{-16}+2q^{-20}+2q^{-22}-q^{-24}+2q^{-26}-4q^{-28}-q^{-30}-3q^{-34}+q^{-36}+q^{-38}$
2,0	$q^{-12}-2q^{-14}-q^{-16}+7q^{-18}-q^{-20}-11q^{-22}+6q^{-24}+16q^{-26}-5q^{-28}-18q^{-30}+9q^{-32}+23q^{-34}-10q^{-36}-16q^{-38}+14q^{-40}+13q^{-42}-5q^{-44}-5q^{-46}+8q^{-48}-2q^{-50}-3q^{-52}+4q^{-54}-6q^{-56}-14q^{-58}+2q^{-60}+11q^{-62}-15q^{-64}-15q^{-66}+9q^{-68}+15q^{-70}-9q^{-72}-14q^{-74}+13q^{-76}+14q^{-78}-3q^{-80}-8q^{-82}+3q^{-84}+8q^{-86}-2q^{-88}-5q^{-90}-2q^{-92}+q^{-94}+q^{-96}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-12}-2q^{-14}+6q^{-18}-7q^{-20}-3q^{-22}+18q^{-24}-10q^{-26}-11q^{-28}+28q^{-30}-8q^{-32}-16q^{-34}+31q^{-36}-3q^{-38}-11q^{-40}+14q^{-42}+2q^{-44}-9q^{-46}-13q^{-48}+4q^{-50}+q^{-52}-25q^{-54}+7q^{-56}+19q^{-58}-23q^{-60}+8q^{-62}+20q^{-64}-20q^{-66}+6q^{-68}+10q^{-70}-12q^{-72}+4q^{-74}+2q^{-76}-3q^{-78}+q^{-80}$
1,0,0	$q^{-9}-2q^{-11}+2q^{-13}-q^{-15}+2q^{-17}-2q^{-19}+4q^{-21}+2q^{-25}+2q^{-27}+2q^{-29}+2q^{-31}-q^{-33}+2q^{-35}-4q^{-37}-q^{-39}-4q^{-41}-3q^{-45}+q^{-47}+q^{-49}+q^{-51}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-12}-2q^{-14}+4q^{-16}-8q^{-18}+13q^{-20}-17q^{-22}+24q^{-24}-28q^{-26}+33q^{-28}-32q^{-30}+28q^{-32}-16q^{-34}+5q^{-36}+13q^{-38}-27q^{-40}+44q^{-42}-56q^{-44}+63q^{-46}-65q^{-48}+58q^{-50}-49q^{-52}+33q^{-54}-17q^{-56}-q^{-58}+15q^{-60}-26q^{-62}+32q^{-64}-34q^{-66}+32q^{-68}-28q^{-70}+20q^{-72}-14q^{-74}+8q^{-76}-3q^{-78}+q^{-80}$
1,0	$q^{-18}-2q^{-22}-2q^{-24}+2q^{-26}+7q^{-28}+2q^{-30}-10q^{-32}-10q^{-34}+6q^{-36}+21q^{-38}+7q^{-40}-20q^{-42}-22q^{-44}+11q^{-46}+34q^{-48}+7q^{-50}-31q^{-52}-18q^{-54}+23q^{-56}+31q^{-58}-9q^{-60}-28q^{-62}+q^{-64}+29q^{-66}+7q^{-68}-24q^{-70}-13q^{-72}+16q^{-74}+13q^{-76}-17q^{-78}-20q^{-80}+9q^{-82}+20q^{-84}-9q^{-86}-30q^{-88}-4q^{-90}+32q^{-92}+18q^{-94}-26q^{-96}-31q^{-98}+14q^{-100}+37q^{-102}+6q^{-104}-29q^{-106}-18q^{-108}+18q^{-110}+22q^{-112}-5q^{-114}-16q^{-116}-4q^{-118}+9q^{-120}+5q^{-122}-3q^{-124}-3q^{-126}+q^{-130}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-30}-2q^{-32}+4q^{-34}-6q^{-36}+6q^{-38}-5q^{-40}+11q^{-44}-21q^{-46}+33q^{-48}-38q^{-50}+31q^{-52}-14q^{-54}-15q^{-56}+52q^{-58}-84q^{-60}+107q^{-62}-105q^{-64}+68q^{-66}+3q^{-68}-87q^{-70}+169q^{-72}-206q^{-74}+183q^{-76}-94q^{-78}-40q^{-80}+166q^{-82}-226q^{-84}+203q^{-86}-85q^{-88}-58q^{-90}+169q^{-92}-188q^{-94}+107q^{-96}+45q^{-98}-192q^{-100}+265q^{-102}-220q^{-104}+73q^{-106}+125q^{-108}-289q^{-110}+359q^{-112}-307q^{-114}+147q^{-116}+50q^{-118}-229q^{-120}+322q^{-122}-304q^{-124}+183q^{-126}-14q^{-128}-146q^{-130}+224q^{-132}-204q^{-134}+85q^{-136}+65q^{-138}-188q^{-140}+214q^{-142}-138q^{-144}-16q^{-146}+177q^{-148}-270q^{-150}+259q^{-152}-149q^{-154}-14q^{-156}+158q^{-158}-232q^{-160}+224q^{-162}-139q^{-164}+32q^{-166}+59q^{-168}-106q^{-170}+105q^{-172}-71q^{-174}+33q^{-176}+q^{-178}-19q^{-180}+20q^{-182}-16q^{-184}+8q^{-186}-3q^{-188}+q^{-190}$

.

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["10 101"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$7t^{2}-21t+29-21t^{-1}+7t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$7z^{4}+7z^{2}+1$

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]=

{ 85, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{12}-4q^{11}+7q^{10}-11q^{9}+13q^{8}-14q^{7}+14q^{6}-10q^{5}+7q^{4}-3q^{3}+q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{-4}+3z^{4}a^{-6}+3z^{4}a^{-8}+z^{2}a^{-4}+5z^{2}a^{-6}+5z^{2}a^{-8}-4z^{2}a^{-10}+2a^{-6}+2a^{-8}-4a^{-10}+a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-9}+2z^{9}a^{-11}+6z^{8}a^{-8}+11z^{8}a^{-10}+5z^{8}a^{-12}+7z^{7}a^{-7}+10z^{7}a^{-9}+7z^{7}a^{-11}+4z^{7}a^{-13}+6z^{6}a^{-6}-6z^{6}a^{-8}-24z^{6}a^{-10}-11z^{6}a^{-12}+z^{6}a^{-14}+3z^{5}a^{-5}-8z^{5}a^{-7}-31z^{5}a^{-9}-31z^{5}a^{-11}-11z^{5}a^{-13}+z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}+15z^{4}a^{-10}+3z^{4}a^{-12}-2z^{4}a^{-14}-2z^{3}a^{-5}+4z^{3}a^{-7}+26z^{3}a^{-9}+28z^{3}a^{-11}+8z^{3}a^{-13}-z^{2}a^{-4}+7z^{2}a^{-6}-z^{2}a^{-8}-9z^{2}a^{-10}+z^{2}a^{-12}+z^{2}a^{-14}-8za^{-9}-9za^{-11}-za^{-13}-2a^{-6}+2a^{-8}+4a^{-10}+a^{-12}$

Vassiliev invariants

V2 and V3:

(7, 17)

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 $28$ $136$ $392$ ${\frac {2642}{3}}$ ${\frac {406}{3}}$ $3808$ ${\frac {18928}{3}}$ ${\frac {3328}{3}}$ $808$ ${\frac {10976}{3}}$ $9248$ ${\frac {73976}{3}}$ ${\frac {11368}{3}}$ ${\frac {1387657}{30}}$ ${\frac {25286}{15}}$ ${\frac {778634}{45}}$ ${\frac {4631}{18}}$ ${\frac {67657}{30}}$

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 10 101. 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                   3   -3 21                 4 1   3 19               7 3     -4 17             6 4       2 15           8 7         -1 13         6 6           0 11       4 8             4 9     3 6               -3 7     4                 4 5 1 3                   -2 3 1                     1

Integral Khovanov Homology (db, data source)

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

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, 101]]
Out[2]=	10
In[3]:=	PD[Knot[10, 101]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 16, 1, 15], X[16, 12, 17, 11], X[12, 20, 13, 19], X[18, 8, 19, 7], X[6, 14, 7, 13], X[8, 18, 9, 17], X[2, 10, 3, 9]]
In[4]:=	GaussCode[Knot[10, 101]]
Out[4]=	GaussCode[1, -10, 2, -1, 3, -8, 7, -9, 10, -2, 5, -6, 8, -3, 4, -5, 9, -7, 6, -4]
In[5]:=	BR[Knot[10, 101]]
Out[5]=	BR[5, {1, 1, 1, 2, -1, 3, -2, 1, 3, 2, 2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[10, 101]][t]
Out[6]=	7 21 2 29 + -- - -- - 21 t + 7 t 2 t t
In[7]:=	Conway[Knot[10, 101]][z]
Out[7]=	2 4 1 + 7 z + 7 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 101], Knot[11, Alternating, 200]}
In[9]:=	{KnotDet[Knot[10, 101]], KnotSignature[Knot[10, 101]]}
Out[9]=	{85, 4}
In[10]:=	J=Jones[Knot[10, 101]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 q - 3 q + 7 q - 10 q + 14 q - 14 q + 13 q - 11 q + 7 q - 11 12 4 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 101]}
In[12]:=	A2Invariant[Knot[10, 101]][q]
Out[12]=	6 8 10 12 14 16 20 22 24 26 q - 2 q + 2 q + q - 2 q + 4 q + 2 q + 2 q - q + 2 q - 28 30 34 36 38 4 q - q - 3 q + q + q
In[13]:=	Kauffman[Knot[10, 101]][a, z]
Out[13]=	2 2 2 2 2 -12 4 2 2 z 9 z 8 z z z 9 z z 7 z a + --- + -- - -- - --- - --- - --- + --- + --- - ---- - -- + ---- - 10 8 6 13 11 9 14 12 10 8 6 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 z 8 z 28 z 26 z 4 z 2 z 2 z 3 z 15 z z -- + ---- + ----- + ----- + ---- - ---- - ---- + ---- + ----- + -- - 4 13 11 9 7 5 14 12 10 8 a a a a a a a a a a 4 4 5 5 5 5 5 6 6 8 z z 11 z 31 z 31 z 8 z 3 z z 11 z ---- + -- - ----- - ----- - ----- - ---- + ---- + --- - ----- - 6 4 13 11 9 7 5 14 12 a a a a a a a a a 6 6 6 7 7 7 7 8 8 24 z 6 z 6 z 4 z 7 z 10 z 7 z 5 z 11 z ----- - ---- + ---- + ---- + ---- + ----- + ---- + ---- + ----- + 10 8 6 13 11 9 7 12 10 a a a a a a a a a 8 9 9 6 z 2 z 2 z ---- + ---- + ---- 8 11 9 a a a
In[14]:=	{Vassiliev[2][Knot[10, 101]], Vassiliev[3][Knot[10, 101]]}
Out[14]=	{0, 17}
In[15]:=	Kh[Knot[10, 101]][q, t]
Out[15]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 3 q t + 4 q t + 3 q t + 6 q t + 4 q t + 8 q t + 13 4 13 5 15 5 15 6 17 6 17 7 6 q t + 6 q t + 8 q t + 7 q t + 6 q t + 4 q t + 19 7 19 8 21 8 21 9 23 9 25 10 7 q t + 3 q t + 4 q t + q t + 3 q t + q t