9 48

9_47

9_49

Visit 9 48's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 48's page at Knotilus! Visit 9 48's page at the original Knot Atlas! 9 48 Quick Notes

9 48 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X12,8,13,7 X3,11,4,10 X11,3,12,2 X14,6,15,5 X6,14,7,13 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code	-1, 4, -3, 1, 5, -6, 2, 9, -8, 3, -4, -2, 6, -5, -7, 8, -9, 7
Dowker-Thistlethwaite code	4 10 -14 -12 16 2 -6 18 8
Conway Notation	[21,21,21-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 2 Maximal Thurston-Bennequin number [-1][-8] Hyperbolic Volume 9.53188 A-Polynomial See Data:9 48/A-polynomial

[edit Notes for 9 48's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 2

[edit Notes for 9 48's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+7t-11+7t^{-1}-t^{-2}$
Conway polynomial	$-z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 27, 2 }
Jones polynomial	$-2q^{6}+3q^{5}-4q^{4}+6q^{3}-4q^{2}+4q-3+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-z^{2}a^{-2}+3z^{2}a^{-4}+z^{2}+3a^{-4}-2a^{-6}$
Kauffman polynomial (db, data sources)	$z^{7}a^{-3}+z^{7}a^{-5}+3z^{6}a^{-2}+4z^{6}a^{-4}+z^{6}a^{-6}+3z^{5}a^{-1}+2z^{5}a^{-3}-z^{5}a^{-5}-5z^{4}a^{-2}-6z^{4}a^{-4}+z^{4}-5z^{3}a^{-1}-3z^{3}a^{-3}+5z^{3}a^{-5}+3z^{3}a^{-7}+2z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}-z^{2}-za^{-3}-5za^{-5}-4za^{-7}+3a^{-4}+2a^{-6}$
The A2 invariant	$q^{4}-q^{2}-1+q^{-2}-q^{-4}+2q^{-6}+q^{-8}+2q^{-10}+2q^{-12}+q^{-16}-2q^{-18}-2q^{-20}$
The G2 invariant	$q^{18}-2q^{16}+4q^{14}-6q^{12}+3q^{10}+q^{8}-6q^{6}+14q^{4}-14q^{2}+15-8q^{-2}-7q^{-4}+16q^{-6}-23q^{-8}+18q^{-10}-9q^{-12}-5q^{-14}+13q^{-16}-13q^{-18}+10q^{-20}+q^{-22}-13q^{-24}+16q^{-26}-13q^{-28}+q^{-30}+15q^{-32}-24q^{-34}+27q^{-36}-14q^{-38}+11q^{-40}+7q^{-42}-22q^{-44}+29q^{-46}-22q^{-48}+19q^{-50}-2q^{-52}-14q^{-54}+21q^{-56}-8q^{-58}+6q^{-60}+q^{-62}-15q^{-64}+14q^{-66}-5q^{-68}-8q^{-70}+14q^{-72}-24q^{-74}+21q^{-76}-5q^{-78}-10q^{-80}+11q^{-82}-18q^{-84}+16q^{-86}-9q^{-88}-2q^{-90}+3q^{-92}-7q^{-94}+7q^{-96}-2q^{-98}+q^{-100}+q^{-102}$

Further Quantum Invariants

Further quantum knot invariants for 9_48.

A1 Invariants.

Weight	Invariant
1	$q^{3}-2q+q^{-1}+2q^{-5}+2q^{-7}-q^{-9}+q^{-11}-2q^{-13}$
2	$q^{10}-2q^{8}-2q^{6}+6q^{4}-6+5q^{-2}+2q^{-4}-7q^{-6}+2q^{-8}+4q^{-10}-2q^{-12}+q^{-14}+4q^{-16}+4q^{-18}-5q^{-20}+6q^{-24}-7q^{-26}-4q^{-28}+5q^{-30}-3q^{-32}-3q^{-34}+3q^{-36}+q^{-38}$
3	$q^{21}-2q^{19}-2q^{17}+3q^{15}+6q^{13}-13q^{9}-4q^{7}+14q^{5}+11q^{3}-13q-19q^{-1}+12q^{-3}+26q^{-5}-4q^{-7}-27q^{-9}-q^{-11}+23q^{-13}+7q^{-15}-22q^{-17}-7q^{-19}+14q^{-21}+11q^{-23}-7q^{-25}-9q^{-27}+2q^{-29}+14q^{-31}+9q^{-33}-15q^{-35}-12q^{-37}+12q^{-39}+21q^{-41}-15q^{-43}-27q^{-45}+4q^{-47}+25q^{-49}-2q^{-51}-24q^{-53}-6q^{-55}+19q^{-57}+9q^{-59}-9q^{-61}-7q^{-63}+5q^{-65}+8q^{-67}-q^{-69}-2q^{-71}-2q^{-73}$
4	$q^{36}-2q^{34}-2q^{32}+3q^{30}+3q^{28}+6q^{26}-7q^{24}-13q^{22}-4q^{20}+5q^{18}+32q^{16}+8q^{14}-25q^{12}-34q^{10}-19q^{8}+52q^{6}+51q^{4}-63-81q^{-2}+30q^{-4}+93q^{-6}+64q^{-8}-44q^{-10}-120q^{-12}-27q^{-14}+78q^{-16}+104q^{-18}+2q^{-20}-106q^{-22}-62q^{-24}+33q^{-26}+90q^{-28}+31q^{-30}-60q^{-32}-57q^{-34}-q^{-36}+53q^{-38}+38q^{-40}-14q^{-42}-43q^{-44}-28q^{-46}+20q^{-48}+50q^{-50}+35q^{-52}-42q^{-54}-62q^{-56}-14q^{-58}+60q^{-60}+82q^{-62}-29q^{-64}-94q^{-66}-63q^{-68}+48q^{-70}+120q^{-72}+14q^{-74}-85q^{-76}-97q^{-78}+5q^{-80}+110q^{-82}+57q^{-84}-27q^{-86}-84q^{-88}-39q^{-90}+52q^{-92}+55q^{-94}+21q^{-96}-34q^{-98}-40q^{-100}+q^{-102}+17q^{-104}+20q^{-106}-q^{-108}-13q^{-110}-7q^{-112}-3q^{-114}+3q^{-116}+3q^{-118}+q^{-120}$
5	$q^{55}-2q^{53}-2q^{51}+3q^{49}+3q^{47}+3q^{45}-q^{43}-7q^{41}-13q^{39}-4q^{37}+14q^{35}+23q^{33}+20q^{31}-4q^{29}-37q^{27}-57q^{25}-18q^{23}+48q^{21}+88q^{19}+69q^{17}-24q^{15}-128q^{13}-149q^{11}-29q^{9}+144q^{7}+229q^{5}+139q^{3}-105q-305q^{-1}-269q^{-3}+18q^{-5}+328q^{-7}+385q^{-9}+115q^{-11}-291q^{-13}-471q^{-15}-249q^{-17}+206q^{-19}+491q^{-21}+361q^{-23}-85q^{-25}-460q^{-27}-411q^{-29}-18q^{-31}+372q^{-33}+424q^{-35}+98q^{-37}-283q^{-39}-372q^{-41}-141q^{-43}+180q^{-45}+312q^{-47}+159q^{-49}-111q^{-51}-235q^{-53}-155q^{-55}+37q^{-57}+179q^{-59}+157q^{-61}+10q^{-63}-131q^{-65}-166q^{-67}-64q^{-69}+96q^{-71}+196q^{-73}+133q^{-75}-74q^{-77}-229q^{-79}-196q^{-81}+25q^{-83}+265q^{-85}+295q^{-87}+19q^{-89}-300q^{-91}-372q^{-93}-115q^{-95}+286q^{-97}+454q^{-99}+203q^{-101}-245q^{-103}-491q^{-105}-306q^{-107}+150q^{-109}+483q^{-111}+382q^{-113}-30q^{-115}-399q^{-117}-420q^{-119}-84q^{-121}+288q^{-123}+395q^{-125}+182q^{-127}-139q^{-129}-317q^{-131}-223q^{-133}+23q^{-135}+206q^{-137}+204q^{-139}+57q^{-141}-98q^{-143}-153q^{-145}-90q^{-147}+23q^{-149}+79q^{-151}+67q^{-153}+19q^{-155}-27q^{-157}-45q^{-159}-21q^{-161}+3q^{-163}+12q^{-165}+17q^{-167}+8q^{-169}-q^{-171}-4q^{-173}-2q^{-175}-2q^{-177}$

A2 Invariants.

Weight	Invariant
1,0	$q^{4}-q^{2}-1+q^{-2}-q^{-4}+2q^{-6}+q^{-8}+2q^{-10}+2q^{-12}+q^{-16}-2q^{-18}-2q^{-20}$
1,1	$q^{12}-4q^{10}+10q^{8}-20q^{6}+30q^{4}-42q^{2}+58-62q^{-2}+59q^{-4}-48q^{-6}+26q^{-8}-2q^{-10}-34q^{-12}+68q^{-14}-86q^{-16}+112q^{-18}-108q^{-20}+114q^{-22}-94q^{-24}+70q^{-26}-40q^{-28}+2q^{-30}+22q^{-32}-46q^{-34}+61q^{-36}-68q^{-38}+56q^{-40}-48q^{-42}+31q^{-44}-22q^{-46}+10q^{-48}-2q^{-50}+2q^{-52}+2q^{-54}$
2,0	$q^{12}-q^{10}-2q^{8}+3q^{4}+3q^{2}-4-q^{-2}+4q^{-4}+q^{-6}-4q^{-8}-2q^{-10}+2q^{-12}-q^{-14}+2q^{-18}+5q^{-20}+2q^{-22}+6q^{-24}+4q^{-26}-q^{-28}+q^{-30}+3q^{-32}-2q^{-34}-7q^{-36}-4q^{-38}-q^{-40}-4q^{-42}-6q^{-44}+4q^{-48}+4q^{-50}+q^{-52}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{8}-2q^{6}+3q^{2}-4+3q^{-2}+4q^{-4}-6q^{-6}+q^{-10}-4q^{-12}+6q^{-16}+6q^{-18}+7q^{-20}+4q^{-22}+6q^{-24}-3q^{-26}-9q^{-28}-q^{-30}-6q^{-32}-7q^{-34}+4q^{-36}+q^{-38}-q^{-40}+3q^{-42}$
1,0,0	$q^{5}-q^{3}-q^{-1}+q^{-3}-q^{-5}+q^{-7}+q^{-9}+q^{-11}+2q^{-13}+2q^{-15}+3q^{-17}+q^{-21}-2q^{-23}-2q^{-25}-2q^{-27}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{10}-q^{8}-2q^{6}+2q^{4}+q^{2}-2+5q^{-4}+q^{-6}-5q^{-8}+3q^{-12}-5q^{-14}-7q^{-16}+3q^{-18}+q^{-20}+q^{-22}+9q^{-24}+14q^{-26}+9q^{-28}+9q^{-30}+10q^{-32}+q^{-34}-10q^{-36}-8q^{-38}-7q^{-40}-13q^{-42}-10q^{-44}-q^{-46}+2q^{-48}+2q^{-52}+4q^{-54}+3q^{-56}$
1,0,0,0	$q^{6}-q^{4}-q^{-2}+q^{-4}-q^{-6}+q^{-8}+q^{-12}+q^{-14}+2q^{-16}+2q^{-18}+3q^{-20}+3q^{-22}+q^{-26}-2q^{-28}-2q^{-30}-2q^{-32}-2q^{-34}$

B2 Invariants.

Weight	Invariant
0,1	$q^{8}-2q^{6}+4q^{4}-5q^{2}+4-5q^{-2}+4q^{-4}-2q^{-6}+3q^{-10}-4q^{-12}+8q^{-14}-8q^{-16}+10q^{-18}-7q^{-20}+8q^{-22}-4q^{-24}+3q^{-26}+q^{-28}-3q^{-30}+4q^{-32}-5q^{-34}+4q^{-36}-5q^{-38}+3q^{-40}-3q^{-42}$
1,0	$q^{14}-2q^{10}-2q^{8}+2q^{6}+4q^{4}-q^{2}-4-q^{-2}+6q^{-4}+4q^{-6}-4q^{-8}-5q^{-10}+3q^{-14}-q^{-16}-4q^{-18}-2q^{-20}+4q^{-22}+3q^{-24}+6q^{-30}+7q^{-32}+3q^{-34}-2q^{-36}+2q^{-38}+4q^{-40}-7q^{-44}-5q^{-46}+q^{-48}+q^{-50}-5q^{-52}-7q^{-54}-q^{-56}+4q^{-58}+3q^{-60}-2q^{-62}-2q^{-64}+q^{-66}+3q^{-68}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{10}-2q^{8}+2q^{6}-3q^{4}+4q^{2}-4+4q^{-2}-3q^{-4}+4q^{-6}-2q^{-8}-q^{-10}-2q^{-14}+2q^{-16}-6q^{-18}+6q^{-20}-3q^{-22}+12q^{-24}-q^{-26}+12q^{-28}+10q^{-32}-q^{-34}-5q^{-38}-6q^{-40}-2q^{-42}-8q^{-44}-q^{-46}-6q^{-48}+4q^{-50}-2q^{-52}+4q^{-54}-2q^{-56}+3q^{-58}$

G2 Invariants.

Weight

Invariant

1,0

$q^{18}-2q^{16}+4q^{14}-6q^{12}+3q^{10}+q^{8}-6q^{6}+14q^{4}-14q^{2}+15-8q^{-2}-7q^{-4}+16q^{-6}-23q^{-8}+18q^{-10}-9q^{-12}-5q^{-14}+13q^{-16}-13q^{-18}+10q^{-20}+q^{-22}-13q^{-24}+16q^{-26}-13q^{-28}+q^{-30}+15q^{-32}-24q^{-34}+27q^{-36}-14q^{-38}+11q^{-40}+7q^{-42}-22q^{-44}+29q^{-46}-22q^{-48}+19q^{-50}-2q^{-52}-14q^{-54}+21q^{-56}-8q^{-58}+6q^{-60}+q^{-62}-15q^{-64}+14q^{-66}-5q^{-68}-8q^{-70}+14q^{-72}-24q^{-74}+21q^{-76}-5q^{-78}-10q^{-80}+11q^{-82}-18q^{-84}+16q^{-86}-9q^{-88}-2q^{-90}+3q^{-92}-7q^{-94}+7q^{-96}-2q^{-98}+q^{-100}+q^{-102}$

.

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["9 48"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{2}+7t-11+7t^{-1}-t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{4}+3z^{2}+1$

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-2q^{6}+3q^{5}-4q^{4}+6q^{3}-4q^{2}+4q-3+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{-2}-z^{2}a^{-2}+3z^{2}a^{-4}+z^{2}+3a^{-4}-2a^{-6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{-3}+z^{7}a^{-5}+3z^{6}a^{-2}+4z^{6}a^{-4}+z^{6}a^{-6}+3z^{5}a^{-1}+2z^{5}a^{-3}-z^{5}a^{-5}-5z^{4}a^{-2}-6z^{4}a^{-4}+z^{4}-5z^{3}a^{-1}-3z^{3}a^{-3}+5z^{3}a^{-5}+3z^{3}a^{-7}+2z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}-z^{2}-za^{-3}-5za^{-5}-4za^{-7}+3a^{-4}+2a^{-6}$

Vassiliev invariants

V2 and V3:

(3, 5)

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 $12$ $40$ $72$ $190$ $42$ $480$ ${\frac {2800}{3}}$ ${\frac {448}{3}}$ $200$ $288$ $800$ $2280$ $504$ ${\frac {46031}{10}}$ $-{\frac {6226}{15}}$ ${\frac {38422}{15}}$ ${\frac {337}{6}}$ ${\frac {4111}{10}}$

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=$2 is the signature of 9 48. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 χ 13               2 -2 11             1   1 9           3 2   -1 7         3 1     2 5       1 3       2 3     3 3         0 1   1 2           1 -1   2             -2 -3 1               1

Integral Khovanov Homology (db, data source)

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

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[9, 48]]
Out[2]=	9
In[3]:=	PD[Knot[9, 48]]
Out[3]=	PD[X[1, 4, 2, 5], X[12, 8, 13, 7], X[3, 11, 4, 10], X[11, 3, 12, 2], X[14, 6, 15, 5], X[6, 14, 7, 13], X[15, 18, 16, 1], X[9, 17, 10, 16], X[17, 9, 18, 8]]
In[4]:=	GaussCode[Knot[9, 48]]
Out[4]=	GaussCode[-1, 4, -3, 1, 5, -6, 2, 9, -8, 3, -4, -2, 6, -5, -7, 8, -9, 7]
In[5]:=	BR[Knot[9, 48]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, 1, -3, 2, -1, 2, -3}]
In[6]:=	alex = Alexander[Knot[9, 48]][t]
Out[6]=	-2 7 2 -11 - t + - + 7 t - t t
In[7]:=	Conway[Knot[9, 48]][z]
Out[7]=	2 4 1 + 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 48], Knot[11, NonAlternating, 1]}
In[9]:=	{KnotDet[Knot[9, 48]], KnotSignature[Knot[9, 48]]}
Out[9]=	{27, 2}
In[10]:=	J=Jones[Knot[9, 48]][q]
Out[10]=	1 2 3 4 5 6 -3 + - + 4 q - 4 q + 6 q - 4 q + 3 q - 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 48]}
In[12]:=	A2Invariant[Knot[9, 48]][q]
Out[12]=	-4 -2 2 4 6 8 10 12 16 18 -1 + q - q + q - q + 2 q + q + 2 q + 2 q + q - 2 q - 20 2 q
In[13]:=	Kauffman[Knot[9, 48]][a, z]
Out[13]=	2 2 2 3 3 3 2 3 4 z 5 z z 2 z 2 z 2 z 3 z 5 z 3 z -- + -- - --- - --- - -- - z - -- + ---- + ---- + ---- + ---- - ---- - 6 4 7 5 3 6 4 2 7 5 3 a a a a a a a a a a a 3 4 4 5 5 5 6 6 6 7 5 z 4 6 z 5 z z 2 z 3 z z 4 z 3 z z ---- + z - ---- - ---- - -- + ---- + ---- + -- + ---- + ---- + -- + a 4 2 5 3 a 6 4 2 5 a a a a a a a a 7 z -- 3 a
In[14]:=	{Vassiliev[2][Knot[9, 48]], Vassiliev[3][Knot[9, 48]]}
Out[14]=	{0, 5}
In[15]:=	Kh[Knot[9, 48]][q, t]
Out[15]=	3 1 2 q 3 5 5 2 7 2 2 q + 3 q + ----- + --- + - + 3 q t + q t + 3 q t + 3 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 13 5 q t + 3 q t + 2 q t + q t + 2 q t